#6454866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6362062);