#6483512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470839
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

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

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, 6390708);