Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18704 |
| 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 | |
3150 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)(5,9)(6,8)(7,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,29)(22,27)(23,26)(24,28)(25,31)(30,34)(32,33);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17)(19,21)(20,25)(22,30)(23,31)(24,32)(26,33)(27,29)(28,34);(3,5)(4,9)(6,13)(7,12)(8,15)(10,17)(11,16)(14,18)(20,26)(22,23)(25,33)(27,28)(29,34)(30,31);(5,12)(9,16)(13,18)(15,17)(19,22,24,23)(20,27,28,26)(21,30,32,31)(25,29,34,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255500);