Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#2103 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,14,10,15,11,26)(7,16,18,28,20,31,19,12)(8,21,23,35,24,13,25,30)(17,29,32,36,33,34,22,27);(5,7,11,18,10,20,9,19)(6,12,26,16,15,28,14,31)(8,22,25,17,24,32,23,33)(13,29,35,36,21,34,30,27);(5,8,9,23,10,24,11,25)(6,13,14,30,15,21,26,35)(7,17,18,32,20,33,19,22)(12,27,16,29,28,36,31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266572);