#6491436

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

2879 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)(9,10,14,19,15,20,13,18)(11,21,23,30,25,34,24,16)(12,17,26,31,27,33,28,32)(22,35,37,40,38,39,36,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,29)(18,21)(19,34)(20,30)(26,36)(27,38)(28,37)(31,39)(32,35)(33,40);(2,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,29)(24,34)(25,30)(26,31)(27,33)(28,32)(35,37)(36,39)(38,40);(2,5)(6,8)(9,12,15,27)(10,17,20,33)(11,22,25,38)(13,28,14,26)(16,29,30,40)(18,32,19,31)(21,35,34,39)(23,37,24,36);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,26,27,28)(16,21,30,34)(17,31,33,32)(22,36,38,37)(29,39,40,35)

Polycyclic presentation:

2^9|a^2=f,d^2=i,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, 6398632);