Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10629 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #397 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
760 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);(4,7)(8,10)(11,12)(13,21,24,34)(14,18)(15,19)(16,20)(17,23,31,28)(22,37,25,36)(26,32)(27,33)(29,35)(30,39,41,38)(40,42);(4,8)(6,9)(7,10)(11,13)(12,17)(14,25)(15,28)(16,24)(18,30)(19,34)(20,31)(21,35)(22,27)(23,29)(26,38)(32,37)(33,41)(36,42)(39,40);(3,4)(5,7)(6,10)(8,9)(11,14,15,26,16,27,29,40)(12,18,19,32,20,33,35,42)(13,22,23,38,24,25,28,39)(17,30,21,36,31,41,34,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150108);