#7611867

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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)(3,5)(4,6)(7,8)(9,10)(11,24)(12,18)(13,19)(14,34)(15,22)(16,23)(17,26)(20,33)(21,36)(25,37)(27,29)(28,35)(30,32)(31,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,15,26,16,28,36,27)(10,17,21,24,23,29,22,35)(12,25,30,20,31,34,38,19)(13,32,14,18,33,39,37,40);(2,6)(3,7)(9,12)(10,18)(11,25)(13,29)(14,35)(15,30)(16,31)(17,33)(19,27)(20,26)(21,39)(22,32)(23,40)(24,37)(28,34)(36,38);(2,6)(5,8)(9,13,15,14,16,33,36,37)(10,19,22,34,23,20,21,25)(11,18,27,32,28,40,26,39)(12,29,30,35,31,17,38,24);(9,14,36,13,16,37,15,33)(10,20,22,25,23,19,21,34)(11,18,27,32,28,40,26,39)(12,24,38,17,31,35,30,29)

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519063);