Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2900 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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20,25,22)(21,28,31,34)(23,27,24,32)(26,33,29,30);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,21)(20,26)(22,29)(23,30)(24,33)(25,31)(27,34)(28,32);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22,25,20)(21,29,31,26)(23,32,24,27)(28,33,34,30);(4,11)(9,18)(10,15)(12,17)(19,23,25,24)(20,27,22,32)(21,30,31,33)(26,34,29,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254086);