#6612103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 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)(12,16)(13,15)(17,18,21,25,22,27,20,26)(19,24,28,32,29,31,30,23);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,26)(25,31)(27,32);(2,5)(6,8)(10,15)(11,12)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,32)(27,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,31,32,24);(11,12)(14,16)(17,20,22,21)(18,26,27,25)(19,28,29,30)(23,24,32,31)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519299);