#1337871

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1245067);