Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30964 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,11,13)(9,10,14,12)(15,16)(17,22)(18,27)(19,24)(20,29)(21,25)(23,28)(26,30);(3,4)(5,6)(7,9,11,14)(8,12,13,10)(15,17)(16,22)(18,21)(19,28)(20,26)(23,24)(25,27)(29,30);(4,6)(15,18,21,17)(16,23,25,30)(19,28,20,26)(22,29,27,24);(7,10,11,12)(8,9,13,14)(15,19)(17,26)(18,28)(20,21)(22,27)(24,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409627);