Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#19056 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3549 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3240 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)(5,13)(6,10)(7,11)(8,12)(9,15)(14,17)(16,18)(19,20,24,27)(21,28,30,33)(22,26,23,31)(25,32,34,29);(3,5)(4,9)(6,14)(7,15)(8,16)(10,13)(11,18)(12,17)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(3,5)(4,9)(6,15)(7,14)(8,16)(10,18)(11,13)(12,17)(22,23)(26,31)(28,33)(29,32);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,13,17,18)(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, 6258656);