#5021442

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3042 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)(3,4)(5,6,7,8)(9,10,13,14)(11,17,19,15)(12,16,21,20)(18,23,24,22)(26,27)(29,37)(31,33)(35,40);(10,14)(15,17)(16,20)(22,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(2,3)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38);(1,3,4,2)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4928638);