#6580229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=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, 6487425);