#6456451

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

1789 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)(9,10,13,18,14,19,22,15)(11,20,24,27,21,16,29,30)(12,17,26,32,25,33,36,28)(23,34,37,38,35,31,39,40);(10,15)(11,21)(12,25)(13,22)(16,30)(17,32)(18,19)(20,27)(23,35)(28,33)(31,40)(34,38);(2,5)(6,8)(9,11,22,29,14,21,13,24)(10,16,15,27,19,20,18,30)(12,23,36,39,25,35,26,37)(17,31,28,38,33,34,32,40);(9,12,13,26,14,25,22,36)(10,17,18,32,19,33,15,28)(11,23,24,37,21,35,29,39)(16,31,30,40,20,34,27,38);(1,3,4,7)(2,6,5,8)(11,21)(16,20)(23,35)(24,29)(27,30)(31,34)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=f,e^2=gi,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, 6363647);