Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31213 |
| 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 | |
2242 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)(9,10,13,12)(15,16,17,22,19,24,20,21)(18,26,27,30,28,29,25,23);(3,4)(5,6)(7,9,11,13)(8,12,14,10)(15,17,19,20)(16,21,24,22)(18,27,28,25)(23,29,30,26);(4,6)(7,10,11,12)(8,13,14,9)(16,22)(17,20)(21,24)(23,29)(25,27)(26,30);(8,14)(9,13)(15,18,19,28)(16,23,24,30)(17,25,20,27)(21,26,22,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410980);