Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18340 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2830 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)(5,11)(6,14)(7,9)(8,12)(10,16)(13,18)(15,17)(19,20,24,27)(21,28,30,33)(22,26,23,31)(25,32,34,29);(3,5,7,13)(4,8,9,17)(6,15,16,12)(10,18,14,11)(19,21,24,30)(20,25,27,34)(22,29,23,32)(26,33,31,28);(19,20)(21,25)(22,26)(23,31)(24,27)(28,32)(29,33)(30,34);(4,9)(8,17)(10,14)(11,18)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254085);