Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12)(13,23,28,38)(14,29)(15,19)(16,33)(17,22)(18,26,36,31)(20,24)(21,30)(25,35,40,42)(27,34,41,39)(32,37);(4,7)(8,9)(11,13,24,40,30,41,15,31,17,28,14,25,16,27,32,26)(12,18,19,34,33,42,20,38,22,36,37,39,21,35,29,23);(3,5,6,10)(4,8,7,9)(11,14,30,32,17,24,16,15)(12,19,33,20,22,37,21,29)(13,25,41,26,28,40,27,31)(18,34,42,38,36,39,35,23)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402580);