Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11588 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,14,19,16,20,27,34,15,18,26,31,28,32,39,41)(13,21,23,33,24,35,37,42,22,17,25,29,36,30,38,40);(4,7)(5,10)(8,9)(11,13)(12,17)(14,25)(15,22)(16,24)(18,21)(19,33)(20,30)(23,26)(27,38)(28,36)(29,31)(32,35)(34,42)(37,39)(40,41);(12,18)(13,22)(19,31)(20,32)(23,25)(24,36)(34,41)(37,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388244);