Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17969 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,13)(9,15)(10,14)(12,17)(16,18)(19,20,24,27,25,21,23,29)(22,26,32,33,30,28,31,34);(3,4,7,10)(5,9,11,16)(6,14,13,8)(12,18,17,15)(19,21,25,20)(22,28,30,26)(23,27,24,29)(31,33,32,34);(3,5)(4,9)(6,12)(7,11)(8,15)(10,16)(13,17)(14,18)(19,22,24,32,25,30,23,31)(20,26,27,33,21,28,29,34);(5,11)(9,16)(12,17)(15,18)(20,21)(22,30)(27,29)(31,32)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251993);