#4277846

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1426 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)(10,14)(12,16)(26,30)(27,33)(28,36)(39,40);(9,10)(11,12)(13,14)(15,16)(18,21)(22,24)(25,26,29,30)(27,31,33,34)(28,37,36,32)(35,40,38,39);(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,37)(29,30)(31,33)(32,36)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,11,14,15)(25,27)(26,31)(28,38)(29,33)(30,34)(32,40)(35,36)(37,39);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,28,29,36)(26,32,30,37)(27,35,33,38)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fg,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4185042);