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×#18692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3000 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);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20,24,28)(21,29,32,33)(22,27,23,26)(25,30,34,31);(3,5,8,14)(4,9,11,18)(6,13,7,16)(10,17,15,12)(19,21,24,32)(20,25,28,34)(22,30,23,31)(26,33,27,29);(3,4)(5,9)(6,15)(7,10)(8,11)(12,13)(14,18)(16,17)(20,26)(22,23)(25,33)(27,28)(29,34)(30,31);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(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, 6255485);