Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#21250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,6)(7,8,12,17,13,16,11,25)(9,14,20,29,19,30,26,28)(10,21,24,31,22,35,23,15)(18,32,34,27,33,38,36,37);(3,4,5,6)(7,9,11,20,13,19,12,26)(8,14,17,28,16,30,25,29)(10,18,23,34,22,33,24,36)(15,27,21,32,31,37,35,38);(7,10,11,23,13,22,12,24)(8,15,17,21,16,31,25,35)(9,18,20,34,19,33,26,36)(14,27,28,32,30,37,29,38);(10,22)(15,31)(18,33)(21,35)(23,24)(27,37)(32,38)(34,36)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279224);