#6467851

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,26)(13,18)(14,19)(15,21)(16,20)(17,25)(23,34)(24,32)(27,39)(28,40)(29,35)(30,37)(31,36)(33,38);(9,11,15,25,16,26,14,24)(10,17,19,22,20,32,21,12)(13,28,30,23,31,38,29,39)(18,34,35,40,36,27,37,33);(2,5)(6,8)(9,12,14,32,16,22,15,17)(10,11,19,25,20,26,21,24)(13,27,29,40,31,34,30,33)(18,23,35,38,36,39,37,28);(9,13,14,29,16,31,15,30)(10,18,19,35,20,36,21,37)(11,23,25,38,26,39,24,28)(12,27,32,40,22,34,17,33);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,21)(11,24,26,25)(12,17,22,32)(13,29,31,30)(18,35,36,37)(23,28,39,38)(27,33,34,40)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=hi,e^2=g,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, 6375047);