#7624651

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

1320 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)(9,10,13,18)(11,21,23,38)(12,20,15,17)(14,27,26,34)(16,31,33,24)(19,25,35,29)(22,39,30,36)(28,37,40,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,32)(23,33)(25,34)(26,35)(27,29)(28,36)(30,37)(31,38)(39,40);(9,11,13,23)(10,16,18,33)(12,22,15,30)(14,28,26,40)(17,32,20,37)(19,36,35,39)(21,34,38,27)(24,25,31,29);(2,5)(6,8)(9,12,13,15)(10,17,18,20)(11,22,23,30)(14,25,26,29)(16,32,33,37)(19,34,35,27)(21,36,38,39)(24,28,31,40);(1,3,4,7)(2,6,5,8)(10,18)(16,33)(17,20)(19,35)(21,38)(27,34)(32,37)(36,39)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

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, 7531847);