#6604448

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 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)(3,7)(4,8)(5,6)(17,18)(19,30)(20,32)(21,26)(22,29)(23,28)(24,27)(25,31);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,31)(18,25,28,30)(20,22,21,24)(26,29,32,27);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20)(18,25)(19,24)(21,23)(22,31)(26,29)(27,32)(28,30);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,20)(18,26,28,32)(19,22,31,24)(25,27,30,29);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,22,23,24)(18,27,28,29)(19,20,31,21)(25,32,30,26)

Polycyclic presentation:

2^9|b^2=h,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,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, 6511644);