Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7818 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)(7,8,10,11)(9,13,14,12)(18,26)(21,28)(24,30)(27,29);(8,11)(12,13)(15,16,19,22)(17,23,25,20)(18,21,26,28)(24,29,30,27);(3,4,5,6)(15,16)(17,20)(18,21)(19,22)(23,25)(24,27)(26,28)(29,30);(7,8)(9,12)(10,11)(13,14)(15,17,19,25)(16,20,22,23)(18,24,26,30)(21,27,28,29);(7,9,10,14)(8,12,11,13)(15,18)(16,21)(17,24)(19,26)(20,27)(22,28)(23,29)(25,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541796);