Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7490 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);(6,10)(11,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(5,6,9,10)(7,11,8,12)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,27,21,26);(5,7,9,8)(6,11,10,12)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(5,8,9,7)(6,11,10,12)(15,23)(18,22)(24,28)(26,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483882);