Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29284 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,7)(5,9,10,8)(19,20)(21,28)(22,26)(23,25)(24,29)(27,31)(30,33)(32,34);(4,7)(8,9)(11,12,14,16)(13,17,18,15)(19,21)(20,24)(22,29)(23,30)(25,32)(26,28)(27,33)(31,34);(11,13,14,18)(12,15,16,17)(20,25)(23,31)(24,32)(26,27)(28,33)(30,34);(3,5,6,10)(4,8,7,9)(20,26)(21,29)(25,27)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371154);