Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#22256 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,18)(15,17)(19,20,24,28,25,29,23,33)(21,26,30,34,31,27,22,32);(3,5)(4,9)(6,15)(7,13)(8,14)(10,18)(11,12)(16,17)(19,21,24,30,25,31,23,22)(20,26,28,34,29,27,33,32);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,22,23,31,25,30,24,21)(20,27,28,32,29,26,33,34);(3,7,8,6)(4,10,11,16)(5,13,14,15)(9,18,12,17)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283575);