#72228231

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

At least 8069 conjugacy classes of subgroups; only the 417203 normal subgroups are shown.

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

Polycyclic presentation:

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72228234

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