#6491435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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

2944 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:

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

Polycyclic presentation:

2^9|a^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,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, 6398631);