#6565471

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^d=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472667);