Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#8840 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,5,6,8,10,9)(11,12,17,22)(13,23,14,25)(15,21,16,20)(18,26,19,24);(3,5,7,8,6,9,10,4)(11,13,17,14)(12,18,22,19)(15,26,16,24)(20,25,21,23);(11,14,17,13)(12,19,22,18)(15,26,16,24)(20,25,21,23)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99003);