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×#3390 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | 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 | |
6410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6,9,8)(7,11,12,10)(13,14)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(5,7,9,12)(6,10,8,11)(15,23)(19,27)(22,28)(24,26);(13,15,18,23)(14,19,21,27)(16,24,17,26)(20,28,25,22);(5,8,9,6)(7,10,12,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483388);