#6559663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466859);