#6466326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

1372 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)(19,23)(20,24)(21,25)(22,27)(26,29)(28,32)(30,31);(2,5)(6,8)(17,19,23,18)(20,28,21,31)(22,27,29,26)(24,30,25,32);(9,11,12,16)(10,13,14,15)(17,20,29,32,23,21,22,30)(18,24,26,28,19,25,27,31);(17,21,29,30,23,20,22,32)(18,25,26,31,19,24,27,28);(1,3,4,7)(2,6,5,8)(17,22,23,29)(18,26,19,27)(20,30,21,32)(24,28,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=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, 6373522);