#4779814

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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

1418 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)(3,5,7,8)(9,10)(11,12)(13,14)(15,16)(19,23)(22,24)(25,26,31,34)(27,35,28,32)(29,30,40,33)(36,39,38,37);(10,14)(11,15)(18,21)(22,24)(25,27,31,28)(26,32,34,35)(29,36,40,38)(30,37,33,39);(10,14)(12,16)(17,18)(19,22)(20,21)(23,24)(25,28,31,27)(26,32,34,35)(29,38,40,36)(30,37,33,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,12,14,16)(25,29)(26,33)(27,36)(28,38)(30,34)(31,40)(32,39)(35,37);(1,3,4,7)(2,5,6,8)(9,12)(10,15)(11,14)(13,16)(17,19,20,23)(18,22,21,24)(25,30,31,33)(26,29,34,40)(27,37,28,39)(32,36,35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4687010);