#6269739

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fghi,c^2=gi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6176935);