#6614930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574008
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

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, 6522126);