#7622945

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(2,4)(5,6)(7,17)(8,14)(9,12)(10,27)(11,30)(13,20)(15,29)(16,26)(18,43)(19,41)(21,49)(22,33)(23,52)(24,55)(25,42)(28,40)(31,39)(32,47)(34,48)(35,45)(36,57)(37,54)(38,51)(44,63)(46,67)(50,64)(53,62)(56,61)(58,65)(59,66)(60,68);(1,2,3,4)(6,12)(13,32)(14,33)(15,34)(16,30)(17,41)(27,42)(35,58)(36,40)(37,59)(38,55)(39,49)(43,63)(52,64)(60,62)(61,67);(5,7,9,19)(6,13,12,32)(8,18,22,44)(10,28,25,57)(11,31,26,21)(14,35,33,58)(15,39,34,49)(16,40,30,36)(17,42,41,27)(20,48,47,29)(23,53,50,68)(24,56,51,46)(37,61,59,67)(38,62,55,60)(43,64,63,52)(45,66,65,54);(5,8)(6,14)(7,18)(9,22)(10,23)(11,24)(12,33)(13,35)(15,37)(16,38)(17,43)(19,44)(20,45)(21,46)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,58)(34,59)(36,60)(39,61)(40,62)(41,63)(42,64)(47,65)(48,66)(49,67)(57,68);(8,22)(14,33)(18,44)(23,50)(24,51)(35,58)(37,59)(38,55)(43,63)(45,65)(46,56)(52,64)(53,68)(54,66)(60,62)(61,67)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530141);