#7624894

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1310 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,16)(13,26)(14,28)(15,22)(17,27)(18,25)(20,32)(21,34)(23,40)(24,29)(30,33)(31,37)(35,39)(36,38);(1,3)(2,5)(4,8)(6,7)(9,10,12,16)(11,15,21,30)(13,17,24,32)(14,18,25,28)(19,33,34,22)(20,29,27,26)(23,31,38,35)(36,37,40,39);(9,11,20,37,14,23,13,22,12,21,27,39,25,38,24,33)(10,15,29,40,18,31,17,19,16,30,26,36,28,35,32,34);(3,8)(5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,29)(21,30)(23,31)(24,32)(25,28)(26,27)(33,34)(35,38)(36,39)(37,40);(2,6)(5,7)(10,16)(11,21)(17,32)(18,28)(22,33)(23,38)(26,29)(37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei,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, 7532090);