#6630812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,27)(22,28)(23,29)(24,32)(25,30)(26,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,29)(21,24)(22,26)(23,30)(27,28)(31,32);(17,20,24,31,23,25,22,27)(18,26,28,19,29,21,32,30);(10,14)(13,16)(17,21)(18,27)(19,22)(20,28)(23,26)(24,30)(25,32)(29,31);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,28,29,32)(19,21,30,26)(20,27,25,31)

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538008);