#7611858

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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

1934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519054);