Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9775 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,8)(9,10)(11,12,15,19,17,20,16,26)(13,14,21,23,22,24,25,18);(3,4,8,10,6,9,5,7)(11,13,17,22)(12,18,20,23)(14,19,24,26)(15,25,16,21);(4,9)(7,10)(11,14,16,18,17,24,15,23)(12,13,19,21,20,22,26,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112847);