#6446845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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)(11,14)(12,16)(13,15)(17,18,23,19)(20,28,31,25)(21,26,32,29)(22,30,24,27);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,18)(20,26,31,29)(21,28,32,25)(22,30,24,27);(9,12,13,11)(10,14,15,16)(17,20,23,31)(18,25,19,28)(21,24,32,22)(26,30,29,27);(1,3,4,7)(2,6,5,8)(17,21)(18,26)(19,29)(20,24)(22,31)(23,32)(25,30)(27,28);(17,22,23,24)(18,27,19,30)(20,21,31,32)(25,26,28,29)

Polycyclic presentation:

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

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, 6354041);