#6491497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470231
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3062 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)(9,10,14,18)(11,15,21,30)(12,22)(13,17,27,34)(16,25)(19,28)(20,29,26,33)(23,31)(24,36)(32,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,23)(13,26)(14,21)(16,31)(17,33)(18,30)(19,25)(20,27)(22,28)(24,35)(29,34)(32,39)(36,40)(37,38);(2,5)(6,8)(9,12)(10,16)(11,19)(13,24)(14,25)(15,28)(17,32)(18,22)(20,35)(21,23)(26,37)(27,38)(29,39)(30,31)(33,40)(34,36);(2,5)(6,8)(9,13,14,27)(10,17,18,34)(11,20,21,26)(12,24,25,38)(15,29,30,33)(16,32,22,36)(19,35,23,37)(28,39,31,40);(1,3,4,7)(2,6,5,8)(10,18)(13,27)(15,30)(16,22)(20,26)(24,38)(28,31)(35,37)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=hi,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6398693);