#7624240

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1561 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)(11,18)(12,23)(13,19)(14,27)(15,16)(17,25)(20,29)(21,37)(22,35)(24,39)(26,32)(28,33)(30,36)(31,40)(34,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,16,19,18)(12,21,24,30)(14,22,26,29)(17,31,34,33)(20,32,35,27)(23,36,39,37)(25,28,38,40);(9,12,13,24)(10,17,19,34)(11,21,15,30)(14,28,26,40)(16,31,18,33)(20,36,35,37)(22,38,29,25)(23,32,39,27);(3,8)(5,7)(9,11,13,15)(10,18,19,16)(12,21,24,30)(14,22,26,29)(17,33,34,31)(20,27,35,32)(23,37,39,36)(25,28,38,40);(2,6)(5,7)(10,19)(16,18)(17,34)(20,35)(23,39)(27,32)(31,33)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531436);