#6456445

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

1805 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)(9,10,13,18,14,19,21,15)(11,16,23,28,20,30,34,26)(12,17,25,31,24,32,35,27)(22,29,36,38,33,39,40,37);(10,15)(11,20)(12,24)(13,21)(16,28)(17,31)(18,19)(22,33)(26,30)(27,32)(29,38)(37,39);(2,5)(6,8)(9,11,21,34,14,20,13,23)(10,16,15,26,19,30,18,28)(12,22,35,40,24,33,25,36)(17,29,27,37,32,39,31,38);(9,12,13,25,14,24,21,35)(10,17,18,31,19,32,15,27)(11,22,23,36,20,33,34,40)(16,29,28,38,30,39,26,37);(1,3,4,7)(2,6,5,8)(11,20)(16,30)(22,33)(23,34)(26,28)(29,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=f,e^2=g,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, 6363641);