Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18984 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12)(13,21)(14,23)(15,18)(16,19)(17,20)(22,25)(24,26);(5,9)(7,10)(11,13)(12,14)(15,22)(16,23)(17,24)(18,21)(19,26)(20,25);(3,4)(5,10)(6,8)(7,9)(11,14)(12,13)(15,21)(16,26)(17,25)(18,22)(19,23)(20,24);(11,15,17,16)(12,18,20,19)(13,22,24,23)(14,21,25,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258559);