#6794979

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 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)(9,10)(11,15)(12,14)(13,16)(17,18,21,26,22,27,29,24)(19,23,30,25,28,32,20,31);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,31)(25,26)(27,32);(11,16)(13,15)(18,24)(19,28)(21,29)(23,25)(26,27)(31,32);(2,5)(6,8)(10,14)(13,15)(17,20,22,30)(18,25,27,31)(19,29,28,21)(23,26,32,24);(1,3,4,7)(2,6,5,8)(19,28)(20,30)(23,32)(25,31)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6702175);