#6813517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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	#39909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6720713);