Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1611 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(6,9)(7,11)(10,12)(13,14,17,20)(15,18,22,25)(16,19,23,26)(21,27,28,24);(5,6,8,10)(7,12,11,9)(13,15)(14,18)(16,21)(17,22)(19,24)(20,25)(23,28)(26,27);(13,16,17,23)(14,19,20,26)(15,21,22,28)(18,24,25,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116050);