#6543431

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450627);