Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60815 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(12,17)(13,21)(14,16)(18,25)(22,26)(23,24);(3,5,6,10)(4,7,8,9)(11,12,15,17)(13,22,23,18)(14,19,16,20)(21,25,24,26);(5,10)(7,9)(11,13)(12,18)(14,21)(15,23)(16,24)(17,22)(19,26)(20,25);(4,8)(7,9)(11,14,15,16)(12,19,17,20)(13,21,23,24)(18,26,22,25);(12,17)(18,22)(19,20)(25,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482716);