#6560577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#39386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(19,25)(20,28)(22,29)(24,30)(26,31)(27,32);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,26,27,22)(20,24,28,30)(25,31,32,29);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19,20,25,21,27,28,32)(18,22,24,29,23,26,30,31);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(18,23)(22,26)(24,30)(29,31);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(18,23)(22,26)(24,30)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,d^2=f,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=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467773);