#7624234

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,21)(11,19,26,17)(12,28,30,62)(13,23,16,20)(15,35,34,48)(18,42,45,31)(22,33,49,37)(24,52,54,61)(25,40,32,46)(27,59,58,66)(29,63,38,50)(36,51,65,44)(39,55,43,70)(41,57,47,64)(53,71,56,69)(60,68,72,67);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,32)(14,26)(15,27)(16,25)(18,39)(19,21)(20,46)(22,41)(23,40)(28,61)(29,56)(30,54)(31,55)(33,64)(34,58)(35,66)(36,60)(37,57)(38,53)(42,70)(43,45)(44,68)(47,49)(48,59)(50,69)(51,67)(52,62)(63,71)(65,72);(9,12,14,30)(10,18,21,45)(11,24,26,54)(13,29,16,38)(15,36,34,65)(17,39,19,43)(20,44,23,51)(22,50,49,63)(25,53,32,56)(27,60,58,72)(28,48,62,35)(31,33,42,37)(40,67,46,68)(41,69,47,71)(52,66,61,59)(55,64,70,57);(3,8)(5,7)(9,11)(10,19)(12,24)(13,32)(14,26)(15,27)(16,25)(17,21)(18,43)(20,40)(22,47)(23,46)(28,52)(29,56)(30,54)(31,55)(33,64)(34,58)(35,59)(36,60)(37,57)(38,53)(39,45)(41,49)(42,70)(44,67)(48,66)(50,71)(51,68)(61,62)(63,69)(65,72);(2,6)(5,7)(9,13,14,16)(10,20,21,23)(11,25,26,32)(12,29,30,38)(15,33,34,37)(17,40,19,46)(18,44,45,51)(22,48,49,35)(24,53,54,56)(27,57,58,64)(28,50,62,63)(31,36,42,65)(39,67,43,68)(41,66,47,59)(52,69,61,71)(55,72,70,60)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531430);