#6491431

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

2899 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)(9,10,15,26,16,12,14,19)(11,20,30,17,24,32,23,21)(13,18,29,38,27,25,28,33)(22,34,39,31,36,40,37,35);(1,2)(3,6)(4,5)(7,8)(9,11,16,24)(10,17,12,21)(13,22,27,36)(14,30,15,23)(18,31,25,35)(19,32,26,20)(28,39,29,37)(33,40,38,34);(2,5)(6,8)(9,12)(10,16)(11,21)(13,25)(14,26)(15,19)(17,24)(18,27)(20,30)(22,35)(23,32)(28,38)(29,33)(31,36)(34,39)(37,40);(2,5)(6,8)(9,13)(10,18)(11,22)(12,25)(14,28)(15,29)(16,27)(17,31)(19,33)(20,34)(21,35)(23,37)(24,36)(26,38)(30,39)(32,40);(1,3,4,7)(2,6,5,8)(13,27)(18,25)(22,36)(28,29)(31,35)(33,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=h,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, 6398627);