Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#39855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(15,16,19,22)(17,23,25,20)(18,21,27,30)(24,29,26,28);(8,12)(10,13)(15,17,19,25)(16,20,22,23)(18,26,27,24)(21,29,30,28);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12);(7,9,11,14)(8,13,12,10)(15,16)(17,20)(18,21)(19,22)(23,25)(24,28)(26,29)(27,30);(7,10,11,13)(8,14,12,9)(15,18,19,27)(16,21,22,30)(17,24,25,26)(20,28,23,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7545038);