#6701931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(9,11,14,10)(12,15,13,16)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,19,21,20)(18,23,24,26)(22,28,29,27)(25,32,31,30);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16);(17,20,21,19)(18,23,24,26)(22,27,29,28)(25,32,31,30)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6609127);