#6554744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461940);