#5752934

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

1647 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,37)(36,40)(38,39);(1,2)(3,7)(4,6)(5,8)(11,15)(13,16)(17,19,20,23)(18,21,22,24)(26,31)(27,34)(33,39)(35,40);(1,2,4,6)(3,5,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,34)(26,32,31,28)(29,35,38,40)(33,37,39,36);(1,2)(3,5)(4,6)(7,8)(17,18)(19,21)(20,22)(23,24)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,33,31,39)(27,35,34,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fhi,d^2=f,e^2=f,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, 5660130);