Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#20749 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,19,16,14,23,18)(13,17,22,25,21,20,26,24);(4,7)(8,10)(11,13)(12,17)(14,20)(15,22)(16,21)(18,24)(19,25)(23,26);(11,14)(12,16)(13,20)(15,18)(17,21)(19,23)(22,24)(25,26);(3,5,6,9)(4,8,7,10)(13,21)(17,20)(22,26)(24,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278192);