#6661114

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(10,14)(11,12)(17,18)(19,23)(20,25)(21,24)(22,26)(27,32)(28,31)(29,30);(3,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,30)(26,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,15,14,16)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,31,32,30);(2,6)(5,8)(18,26)(23,32)(24,25)(30,31)

Polycyclic presentation:

2^9|c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,d^c=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, 6568310);