#6446472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353668);