#6690522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,26,27)(20,32,21,28)(23,29,25,31);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,25,26,23)(19,27,30,24)(28,29,32,31);(2,6)(5,7)(17,21,22,20)(18,25,26,23)(19,28,30,32)(24,31,27,29);(3,8)(5,7)(18,26)(19,29)(23,25)(24,28)(27,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597718);