#7624212

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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

1518 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)(11,19)(12,23)(13,25)(15,28)(17,32)(18,31)(20,35)(21,38)(22,37)(24,27)(26,36)(29,39)(30,40)(33,34);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20)(12,17)(13,18)(14,16)(15,22)(19,35)(21,34)(23,32)(24,39)(25,31)(26,30)(27,29)(28,37)(33,38)(36,40);(9,11,14,22)(10,15,16,20)(12,24,26,38)(13,27,25,21)(17,33,30,39)(18,34,31,29)(19,36,37,23)(28,40,35,32);(3,8)(5,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,28)(20,22)(21,29)(23,32)(24,33)(25,31)(26,30)(27,34)(35,37)(36,40)(38,39);(2,6)(5,7)(10,16)(15,20)(17,30)(18,31)(28,35)(29,34)(32,40)(33,39)

Polycyclic presentation:

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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531408);