#7625026

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

1400 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=f,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,d^c=di,e^c=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, 7532222);