Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25052 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2076 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);(3,4)(5,7)(6,8)(9,10)(11,12)(13,23)(14,27)(15,31)(16,22)(17,21)(18,25)(19,26)(20,28)(24,38)(29,41)(30,36)(32,34)(33,40)(35,42)(37,39);(3,5,6,9)(4,7,8,10)(11,13,17,32,16,26,15,25)(12,18,31,19,22,34,21,23)(14,24,30,42,29,40,28,39)(20,33,41,35,36,38,27,37);(4,8)(7,10)(11,13)(12,19)(14,24)(15,25)(16,26)(17,32)(18,21)(20,35)(22,23)(27,33)(28,39)(29,40)(30,42)(31,34)(36,37)(38,41);(11,14,17,30,16,29,15,28)(12,20,31,41,22,36,21,27)(13,24,32,42,26,40,25,39)(18,33,19,35,34,38,23,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351314);