#1293272

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

1430 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fghi,d^2=g,e^2=gh,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, 1200468);