#7623938

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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
prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531134);