Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#10225 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1620 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)(5,8)(6,9)(7,10)(11,12,17,22)(13,23,26,34)(14,19,30,37)(15,21,16,20)(18,27,33,25)(24,39,40,35)(28,42,41,32)(29,38,31,36);(4,8)(5,6)(9,10)(11,13)(12,18)(14,28)(15,27)(16,25)(17,26)(19,35)(20,34)(21,23)(22,33)(24,29)(30,41)(31,40)(32,36)(37,39)(38,42);(4,9)(5,6)(8,10)(11,14)(12,19)(13,24)(15,29)(16,31)(17,30)(18,32)(20,36)(21,38)(22,37)(23,35)(25,28)(26,40)(27,41)(33,42)(34,39)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141022);