Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(8,10)(11,12)(13,23)(14,27)(15,22)(16,21)(17,20)(18,25)(19,28)(24,32)(26,35)(29,36)(30,33)(31,34)(37,39)(38,41)(40,42);(5,9)(7,10)(11,13)(12,18)(14,24)(15,30)(16,25)(17,26)(19,32)(20,33)(21,35)(22,23)(27,41)(28,39)(29,40)(31,38)(34,42)(36,37);(3,4)(5,10)(6,8)(7,9)(11,14,17,29)(12,19,20,34)(13,24,26,40)(15,28,16,31)(18,32,33,42)(21,27,22,36)(23,37,35,41)(25,38,30,39);(12,20)(18,33)(19,34)(21,22)(23,35)(27,36)(32,42)(37,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258570);