Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#29738 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8)(9,12)(10,11)(13,14)(15,16,20,24,19,23,18,22)(17,25,26,29,27,21,28,30);(3,4)(5,6)(8,11)(12,14)(15,17)(16,21)(18,28)(19,27)(20,26)(22,29)(23,25)(24,30);(4,6)(7,9)(8,12)(10,13)(11,14)(16,22)(18,20)(21,29)(23,24)(25,30)(26,28);(9,13)(12,14)(15,18,19,20)(16,22,23,24)(17,26,27,28)(21,30,25,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340229);