#7624057

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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1285 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 7531253);