#6622411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,25,31)(20,29,32,28)(21,26,22,30);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,25)(26,32)(27,31);(9,10)(11,12)(13,14)(15,16)(17,20,21,31,23,32,22,24)(18,25,30,28,27,19,26,29);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(17,21,23,22)(18,26,27,30)(19,28,25,29)(20,31,32,24);(9,12,13,15)(10,11,14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^d=ehi

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, 6529607);