#6702500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(19,26,29,23)(20,24,21,30)(27,32,28,31);(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,25,26)(20,27,21,28)(24,31,30,32);(2,5)(6,8)(10,14)(11,12)(15,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(1,3,4,7)(2,6,5,8)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32);(9,11)(12,13)(14,15)(19,29)(23,26)(27,28)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6609696);