#6627058

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(9,10)(11,15)(12,16)(13,14)(17,18,23,19)(20,28,31,32)(21,25,22,27)(24,29,26,30);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,23,18)(20,24,31,26)(21,25,22,27)(28,29,32,30);(9,11,13,12)(10,15,14,16)(17,20,21,29,23,31,22,30)(18,24,27,32,19,26,25,28);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,21,23,22)(18,25,19,27)(20,29,31,30)(24,28,26,32);(9,12,13,11)(10,16,14,15)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gh,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, 6534254);