#6466636

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(19,23)(20,25)(21,26)(22,28)(24,27)(29,30)(31,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,18)(20,29,21,32)(22,28,24,27)(25,31,26,30);(9,12,13,11)(10,14,15,16)(17,20,22,30,23,21,24,31)(18,25,28,29,19,26,27,32);(17,21,22,31,23,20,24,30)(18,26,28,32,19,25,27,29);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,27,19,28)(20,30,21,31)(25,32,26,29)

Polycyclic presentation:

2^9|b^2=gi,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, 6373832);