#7611877

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563346
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

1918 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)(11,22)(12,27)(13,20)(14,31)(15,21)(16,18)(17,25)(19,30)(23,35)(24,37)(26,32)(28,36)(29,38)(33,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11,12,25,15,26,16,23)(10,17,18,22,21,35,27,32)(13,29,28,40,30,39,14,24)(19,37,36,33,20,34,31,38);(2,6)(3,7)(9,12,15,16)(10,18,21,27)(11,23,26,25)(13,28,30,14)(17,32,35,22)(19,36,20,31)(24,39,40,29)(33,37,38,34);(2,6)(5,8)(9,13,16,14,15,30,12,28)(10,19,18,36,21,20,27,31)(11,24,25,29,26,40,23,39)(17,33,32,37,35,38,22,34);(9,14,16,30,15,28,12,13)(10,20,27,36,21,19,18,31)(11,24,23,39,26,40,25,29)(17,34,32,33,35,37,22,38)

Polycyclic presentation:

2^9|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,e^d=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, 7519073);