Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1672 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)(5,13)(6,12)(7,9)(8,11)(10,14)(15,18)(16,17)(19,20,24,26)(21,23,28,33)(22,29,32,27)(25,31,34,30);(4,9)(10,17)(11,12)(13,18)(19,21,23,26,24,28,33,20)(22,27,31,34,32,29,30,25);(3,5)(4,10)(6,14)(7,15)(8,16)(9,17)(11,13)(12,18)(19,22)(20,25)(21,27)(23,31)(24,32)(26,34)(28,29)(30,33);(3,6,7,8)(4,11,9,12)(5,14,15,16)(10,13,17,18)(19,23,24,33)(20,21,26,28)(22,30,32,31)(25,29,34,27)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261234);