Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20143 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1748 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);(4,8)(6,14)(9,15)(10,11)(12,18)(16,17)(19,20,24,27)(21,25,23,32)(22,29,31,33)(26,30,34,28);(3,4)(5,9)(6,10)(7,11)(8,14)(12,16)(13,17)(15,18)(19,21,24,23)(20,25,27,32)(22,30,31,28)(26,29,34,33);(3,5,7,13)(4,9,11,17)(6,12,14,18)(8,15,10,16)(19,22)(20,26)(21,28)(23,30)(24,31)(25,33)(27,34)(29,32);(4,10)(6,14)(8,11)(9,16)(12,18)(15,17)(19,21,24,23)(20,25,27,32)(22,28,31,30)(26,33,34,29)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266674);