#6643319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632036
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	#29712	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fhi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efh,c^b=ch,d^b=dfh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550515);