Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22,26,27,25,29,23,30)(24,32,34,36,33,28,31,35);(1,2)(3,4)(5,6,8,9)(7,12,11,10)(13,15,16,20)(14,17,18,19)(21,23,25,26)(22,27,29,30)(24,31,33,34)(28,35,32,36);(2,4)(14,18)(17,19)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,32)(30,36);(5,7,8,11)(6,10,9,12)(15,20)(17,19)(24,33)(28,32)(31,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=dei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292065);