#7623776

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,7,8,5)(11,20)(12,14)(13,25)(16,29)(17,23)(18,33)(21,37)(22,36)(24,27)(26,28)(30,34)(31,39)(32,40)(35,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,23)(13,18)(14,17)(15,19)(20,29)(21,31)(22,32)(24,38)(25,33)(26,34)(27,35)(28,30)(36,40)(37,39);(9,11,15,22)(10,16,19,32)(12,21,14,28)(13,26,27,37)(17,30,23,31)(18,34,35,39)(20,24,36,25)(29,38,40,33);(3,8)(5,7)(9,10)(11,16)(12,23)(13,18)(14,17)(15,19)(20,29)(21,31)(22,32)(24,38)(25,33)(26,34)(27,35)(28,30)(36,40)(37,39);(9,12,15,14)(10,17,19,23)(11,21,22,28)(13,24,27,25)(16,30,32,31)(18,33,35,38)(20,26,36,37)(29,39,40,34)

Polycyclic presentation:

2^9|a^2=f,c^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,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, 7530972);