#1316550

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

1518 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=f,d^2=i,e^2=fgh,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, 1223746);