#7623757

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530953);