#6536561

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	#24390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=gi,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, 6443757);