#6813514

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	C₂×#120	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(17,18,23,22)(19,24,28,31)(20,25,30,21)(26,29,32,27);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,28)(18,24,22,31)(20,29,30,27)(21,26,25,32);(2,6)(5,8)(11,16)(13,15)(17,20,22,21,23,30,18,25)(19,26,24,29,28,32,31,27);(10,14)(13,15)(17,18,23,22)(19,24,28,31)(20,25,30,21)(26,29,32,27);(17,21,22,30,23,25,18,20)(19,27,24,26,28,29,31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fh,d^2=i,e^2=fh,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, 6720710);