Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18943 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3552 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2756 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);(3,4,7,11)(5,9,12,16)(6,8,14,10)(13,15,18,17)(19,20,24,26)(21,28,25,33)(22,30,31,29)(23,32,34,27);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17)(19,21,24,25)(20,22,26,31)(23,30,34,29)(27,33,32,28);(3,5)(4,9)(6,13)(7,12)(8,15)(10,17)(11,16)(14,18)(19,22)(20,25)(21,26)(23,33)(24,31)(27,29)(28,34)(30,32);(5,12)(9,16)(13,18)(15,17)(20,26)(22,31)(27,32)(28,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256113);