Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#9100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1368 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);(3,4,6,9,7,5,8,10)(11,12)(13,21)(14,25,27,18)(15,28)(16,20)(17,23)(19,30)(22,35,38,42)(24,36)(26,33,29,34)(31,37,41,39)(32,40);(3,5,6,10,7,4,8,9)(11,13,15,23,16,24,30,40)(12,17,19,21,20,32,28,36)(14,22,29,39,27,38,26,37)(18,31,34,35,25,41,33,42);(11,14,16,27)(12,18,20,25)(13,22,24,38)(15,29,30,26)(17,31,32,41)(19,34,28,33)(21,35,36,42)(23,39,40,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103428);