Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28759 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2657 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,11,14)(9,13,10,12)(16,20)(17,24)(18,26)(21,28)(25,29)(27,30);(3,4,5,6)(7,9,11,10)(8,12,14,13)(15,16)(17,25)(18,23)(19,20)(21,27)(22,26)(24,28)(29,30);(4,6)(7,9,11,10)(8,13,14,12)(15,17,26,30,19,27,18,24)(16,21,22,28,20,25,23,29);(7,10,11,9)(8,13,14,12)(15,18,19,26)(16,22,20,23)(17,24,27,30)(21,28,25,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374036);