Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#21537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,8,10,9)(11,12)(13,20)(14,18)(15,25)(16,19)(17,23)(21,22)(24,26);(11,13,14,22,16,24,15,23)(12,17,25,26,19,21,18,20);(4,6)(11,12)(13,21)(14,25)(15,18)(16,19)(17,24)(20,22)(23,26);(7,9,10,8)(11,14,16,15)(12,18,19,25)(13,22,24,23)(17,20,21,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280060);