#72272580

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

429921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 64 points:

(1,2)(3,10)(4,12)(5,13)(6,19)(7,35)(8,16)(9,17)(11,24)(14,52)(15,39)(18,54)(20,43)(21,28)(22,58)(23,45)(25,37)(26,48)(27,57)(29,49)(30,34)(31,55)(32,36)(33,51)(38,44)(40,64)(41,42)(46,60)(47,62)(50,63)(53,61)(56,59);(1,3)(2,10)(4,19)(5,20)(6,12)(7,26)(8,23)(9,24)(11,17)(13,43)(14,53)(15,47)(16,45)(18,42)(21,36)(22,40)(25,29)(27,50)(28,32)(30,46)(31,44)(33,59)(34,60)(35,48)(37,49)(38,55)(39,62)(41,54)(51,56)(52,61)(57,63)(58,64);(1,2,8,16)(3,18,23,11)(4,12,28,21)(5,30,33,63)(6,25,36,14)(7,15,39,35)(9,17,41,42)(10,24,45,54)(13,50,51,34)(19,52,32,37)(20,56,59,43)(22,48,38,47)(26,58,62,44)(27,46,60,57)(29,49,53,61)(31,55,64,40);(1,3,8,23)(2,11,16,18)(4,25,28,14)(5,20,33,59)(6,21,36,12)(7,38,39,22)(9,24,41,54)(10,42,45,17)(13,46,51,57)(15,48,35,47)(19,53,32,29)(26,40,62,55)(27,34,60,50)(30,43,63,56)(31,58,64,44)(37,61,52,49);(1,4)(2,12)(3,19)(5,26)(6,10)(7,20)(8,28)(9,29)(11,37)(13,48)(14,54)(15,56)(16,21)(17,49)(18,52)(22,34)(23,32)(24,25)(27,55)(30,58)(31,57)(33,62)(35,43)(36,45)(38,50)(39,59)(40,60)(41,53)(42,61)(44,63)(46,64)(47,51);(1,5)(2,13)(3,20)(4,26)(6,35)(7,19)(8,33)(9,34)(10,43)(11,46)(12,48)(14,55)(15,36)(16,51)(17,30)(18,57)(21,47)(22,29)(23,59)(24,60)(25,40)(27,54)(28,62)(31,52)(32,39)(37,64)(38,53)(41,50)(42,63)(44,61)(45,56)(49,58);(1,6)(2,14)(3,21)(4,11)(5,31)(7,13)(8,36)(9,37)(10,29)(12,23)(15,34)(16,25)(17,32)(18,28)(19,42)(20,58)(22,57)(24,61)(26,56)(27,47)(30,40)(33,64)(35,50)(38,46)(39,51)(41,52)(43,62)(44,59)(45,53)(48,60)(49,54)(55,63);(1,7)(2,15)(3,22)(4,27)(5,32)(6,13)(8,39)(9,40)(10,44)(11,47)(12,46)(14,34)(16,35)(17,31)(18,48)(19,33)(20,53)(21,57)(23,38)(24,26)(25,50)(28,60)(29,59)(30,37)(36,51)(41,55)(42,64)(43,49)(45,58)(52,63)(54,62)(56,61)

Polycyclic presentation:

2^10|c^2=i,d^2=i,c^a=ci,c^b=cij,d^a=dj,d^b=di,d^c=di,e^d=ej,f^c=fj,g^a=gij,g^b=gi,g^c=gi,g^e=gj,g^f=gij,h^a=hi,h^b=hj,h^d=hi,h^e=hij,h^f=hi

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72272585, #72272663
Fitting subgroup of	#72272585, #72272620, #72272647, …
maximal subgroup of	#72272585, #72272620, #72272647
maximal normal subgroup of	#72272585, #72272620, #72272647

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);