Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#7827 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,9)(5,7,10,8)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21)(27,28,31,34)(29,32,36,39)(30,33,37,40)(35,41,42,38);(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23)(27,29,31,36)(28,32,34,39)(30,35,37,42)(33,38,40,41);(4,7)(5,10)(8,9)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)(27,30)(28,33)(29,35)(31,37)(32,38)(34,40)(36,42)(39,41)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72941);