#6446136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
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

1697 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)(9,10,16,22,15,11,14,19)(12,20,28,31,24,17,25,34)(13,18,29,35,27,21,26,32)(23,33,38,39,36,30,37,40);(9,11,15,10)(12,20,24,17)(13,21,27,18)(14,19,16,22)(23,30,36,33)(25,31,28,34)(26,32,29,35)(37,40,38,39);(1,3,4,8)(2,5,6,7)(9,12,16,28,15,24,14,25)(10,17,22,34,11,20,19,31)(13,23,29,38,27,36,26,37)(18,30,35,40,21,33,32,39);(9,13,14,26,15,27,16,29)(10,18,19,32,11,21,22,35)(12,23,25,37,24,36,28,38)(17,30,31,39,20,33,34,40);(12,24)(17,20)(23,36)(25,28)(30,33)(31,34)(37,38)(39,40)

Polycyclic presentation:

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

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, 6353332);