Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#3212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,22)(17,19)(18,21)(20,25)(24,28)(26,27);(5,7,8,12)(6,9,10,11)(14,19)(15,24)(21,22)(25,26);(7,12)(9,11)(13,15)(14,20)(16,25)(17,24)(18,26)(19,27)(21,23)(22,28);(6,10)(9,11)(13,16,17,18)(14,21,19,22)(15,25,24,26)(20,23,27,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482680);