Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60948 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,29)(22,28)(23,25)(24,27)(26,31)(30,34)(32,33);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(20,25)(21,30)(27,28)(31,32);(5,12)(8,11)(13,18)(15,17)(19,21)(20,26)(22,31)(23,30)(24,32)(25,33)(27,29)(28,34);(4,9)(8,11)(10,16)(15,17)(19,22,23,24)(20,27,25,28)(21,31,30,32)(26,29,33,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482982);