Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13245 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,13,19,15,20,14,23)(9,21,24,31,17,11,18,12)(10,25,28,22,30,37,29,33)(16,32,34,38,35,26,36,27);(3,4,5,6)(7,9,21,23,14,24,12,8,15,17,11,19,13,18,31,20)(10,22,32,36,29,37,27,16,30,33,26,34,28,25,38,35);(7,10,11,26,14,29,31,38,15,30,21,32,13,28,12,27)(8,16,9,22,19,34,24,37,20,35,17,33,23,36,18,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402426);