#6558887

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6466083);