Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17986 |
| 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 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1630 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);(3,4,8,12)(5,13,15,18)(6,11,7,10)(9,16,17,14)(19,20,23,27,24,26,32,33)(21,25,29,22,28,30,34,31);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,18)(12,17)(19,21)(20,25)(22,27)(23,29)(24,28)(26,30)(31,33)(32,34);(19,22,23,30,24,31,32,25)(20,21,27,29,26,28,33,34);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,13,17,18)(20,26)(21,28)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252048);