Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×#7986 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,10,8,12,7,11)(5,13,14,9,16,17,15,18)(19,20,23,28,24,27,22,26)(21,29,30,34,31,25,32,33);(3,5,7,15,8,16,6,14)(4,9,10,17,12,18,11,13)(19,21)(20,25)(22,30)(23,32)(24,31)(26,33)(27,29)(28,34);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)(19,22,24,23)(20,26,27,28)(21,30,31,32)(25,33,29,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88455);