#6491473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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

2872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398669);