#6690167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	#36455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,26,29,24)(20,30,21,25)(27,32,31,28);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,24,23,26)(20,27,21,31)(25,32,30,28);(9,12,13,11)(10,14,15,16)(18,23)(19,26)(24,29)(25,30)(27,28)(31,32);(2,6)(5,7)(17,20,22,21)(18,25,23,30)(19,27,29,31)(24,32,26,28);(3,8)(5,7)(17,21,22,20)(18,25,23,30)(19,28,29,32)(24,31,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6597363);