#6604455

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

1397 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)(19,26)(21,22)(24,31)(25,29)(27,32)(28,30);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,23)(21,29)(22,25)(24,28)(26,32)(30,31);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,19)(18,24)(20,27)(21,26)(22,30)(23,28)(25,32)(29,31);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,18)(19,24,28,27)(21,25,22,29)(26,31,30,32);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,21,23,22)(18,25,20,29)(19,26,28,30)(24,32,27,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=f,d^2=fi,e^2=fi,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, 6511651);