Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60399 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C2×#60399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
23392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(8,14)(15,18);(3,4)(5,8)(6,9)(7,10)(11,15)(12,14)(13,16)(17,18);(3,5)(4,8)(6,11)(7,12)(9,15)(10,14)(13,17)(16,18);(3,6)(4,9)(5,11)(7,13)(8,15)(10,16)(12,17)(14,18);(6,13)(9,16)(11,17)(15,18)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482067);