Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19012 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2732 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);(4,9)(6,7)(10,17)(11,14)(12,15)(16,18)(19,20)(21,28)(22,26)(23,31)(24,27)(25,29)(30,33)(32,34);(3,4,8,11)(5,10,13,18)(6,9,7,14)(12,16,15,17)(19,20)(21,25)(22,26)(23,31)(24,27)(28,32)(29,33)(30,34);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,18)(14,17)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(3,6,8,7)(4,9,11,14)(5,12,13,15)(10,16,18,17)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258592);