#6496006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(3,8)(4,6)(5,7)(11,15)(14,16)(18,23)(19,28)(21,24)(25,30)(26,31)(27,32);(9,10)(11,14)(12,13)(15,16)(17,18,24,32,22,27,21,23)(19,25,20,30,28,31,29,26);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,19,22,28)(18,25,27,31)(20,21,29,24)(23,26,32,30);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,29)(18,26,27,30)(19,24,28,21)(23,31,32,25);(9,11,12,15)(10,14,13,16)(19,28)(20,29)(25,31)(26,30)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei

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, 6403202);