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×#18691 |
| 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 | |
3248 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)(5,10)(6,11)(7,16)(8,12)(9,13)(14,17)(15,18)(19,20,24,28)(21,29,32,33)(22,27,23,26)(25,30,34,31);(3,5)(4,9)(6,13)(7,15)(8,14)(10,16)(11,17)(12,18)(19,21)(20,25)(22,30)(23,31)(24,32)(26,33)(27,29)(28,34);(3,5)(4,10)(6,15)(7,13)(8,14)(9,16)(11,18)(12,17)(20,26)(22,23)(25,33)(27,28)(29,34)(30,31);(3,6,8,7)(4,11,12,16)(5,13,14,15)(9,17,18,10)(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, 6255484);