#6573789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4586 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480985);