#6269713

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1424 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,16,12,15)(17,18)(19,24)(20,22)(21,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,2)(3,5)(4,7)(6,8)(10,14)(11,12)(17,18,21,23)(19,22,20,24)(26,30)(27,33)(32,36)(35,39);(2,3)(6,7)(9,11,13,12)(10,15,14,16)(17,19,21,20)(18,22,23,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,15,14,16)(17,20,21,19)(18,22,23,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,22)(21,23)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=gi,d^2=g,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, 6176909);