#1298919

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1582 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)(11,16)(12,14)(17,18,22,24)(19,20,23,21)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(1,2,4,6)(3,5,8,7)(9,10,13,15)(11,14,16,12)(17,19,22,23)(18,20,24,21)(26,27)(31,33)(32,34)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(17,20,22,21)(18,19,24,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,3,4,8)(2,5,6,7)(9,12,13,14)(10,11,15,16)(17,21,22,20)(18,23,24,19)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=f,d^2=i,e^2=g,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, 1206115);