Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22164 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2208 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);(3,4,6,5)(8,11)(9,13)(15,16)(17,22)(18,27)(19,24)(20,29)(21,25)(23,28)(26,30);(3,5,6,4)(7,8,10,11)(9,12,13,14)(15,17,21,18)(16,22,25,27)(19,28,20,26)(23,29,30,24);(8,11)(12,14)(15,18,21,17)(16,23,25,30)(19,28,20,26)(22,29,27,24);(7,9,10,13)(8,12,11,14)(15,19,21,20)(16,24,25,29)(17,26,18,28)(22,30,27,23)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283044);