#72272519

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

429689 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:

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

Polycyclic presentation:

2^10|e^2=j,f^2=ij,g^2=ij,h^2=j,e^a=ej,e^b=ei,e^d=ei,f^a=fij,f^b=fi,f^c=fj,f^e=fj,g^a=gij,g^b=gj,g^d=gj,g^f=gi,h^a=hj,h^b=hij,h^c=hi,h^e=hij,h^g=hj

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72272532, #72272552
Fitting subgroup of	#72272523, #72272532, #72272552, …
maximal subgroup of	#72272523, #72272532, #72272577
maximal normal subgroup of	#72272523, #72272532, #72272577

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, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);