Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30280 |
| 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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2522 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);(7,8,11,14,10,13,9,12)(15,16,19,22)(17,23,25,20)(18,21,26,28)(24,29,30,27);(3,4,5,6)(15,17,19,25)(16,20,22,23)(18,24,26,30)(21,27,28,29);(4,6)(7,8,10,13)(9,14,11,12)(18,26)(21,28)(24,30)(27,29);(15,18,19,26)(16,21,22,28)(17,24,25,30)(20,27,23,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343552);