#4281390

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

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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=fgh,d^2=g,e^2=fghi,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, 4188586);