#7624219

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1537 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)(11,17)(12,18)(13,16)(14,27)(15,21)(19,26)(20,25)(22,31)(23,30)(24,32)(28,36)(29,40)(33,39)(34,38)(35,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,28)(17,18)(19,29)(20,30)(24,36)(25,37)(26,38)(27,35)(31,32)(33,34)(39,40);(10,16)(13,26)(14,25)(15,28)(17,31)(18,32)(19,20)(22,38)(23,37)(24,39)(27,33)(29,30)(34,35)(36,40);(3,8)(5,7)(9,11)(10,17)(12,21)(13,22)(14,23)(15,18)(16,31)(19,34)(20,35)(24,36)(25,37)(26,38)(27,30)(28,32)(29,33)(39,40);(2,6)(5,7)(10,18)(11,21)(16,32)(19,33)(20,27)(22,36)(23,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531415);