#7623779

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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530975);