#5720138

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1714 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5627334);