#7624217

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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

1489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^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,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531413);