#6536560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(17,18,24,21)(19,27,32,28)(20,22,26,23)(25,30,29,31);(9,10)(11,14)(12,13)(15,16)(17,19,23,31,24,32,22,30)(18,25,26,27,21,29,20,28);(2,5)(6,8)(10,13)(14,16)(17,20)(18,23)(19,28)(21,22)(24,26)(25,31)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,24,18)(19,29,32,25)(20,22,26,23)(27,31,28,30);(11,15)(14,16)(17,22,24,23)(18,26,21,20)(19,30,32,31)(25,27,29,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^c=ei,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, 6443756);