#6466302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10,15,19)(11,20,18,14)(12,23,25,16)(13,17,29,27)(21,26,32,30)(22,33,34,28)(24,36,38,31)(35,39,37,40);(9,11,14,10,15,18,20,19)(12,23,26,21,25,16,30,32)(13,27,28,22,29,17,33,34)(24,37,39,36,38,35,40,31);(1,3,4,8)(2,5,6,7)(9,12,20,30,15,25,14,26)(10,16,18,32,19,23,11,21)(13,24,33,40,29,38,28,39)(17,31,34,37,27,36,22,35);(9,13,20,33,15,29,14,28)(10,17,18,34,19,27,11,22)(12,24,30,40,25,38,26,39)(16,31,32,37,23,36,21,35);(12,25)(16,23)(21,32)(24,38)(26,30)(31,36)(35,37)(39,40)

Polycyclic presentation:

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

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, 6373498);