Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,17,21)(13,18,24,34)(14,25,27,35)(15,20,16,29)(19,30,36,26)(22,37,40,42)(23,32,28,33)(31,38,41,39);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,28)(16,23)(17,24)(19,31)(20,33)(21,34)(25,37)(26,39)(27,40)(29,32)(30,38)(35,42)(36,41);(11,14,17,27)(12,19,21,36)(13,22,24,40)(15,26,16,30)(18,31,34,41)(20,35,29,25)(23,38,28,39)(32,37,33,42);(5,10)(7,9)(11,13)(12,18)(14,22)(15,28)(16,23)(17,24)(19,31)(20,33)(21,34)(25,37)(26,39)(27,40)(29,32)(30,38)(35,42)(36,41);(11,15,17,16)(12,20,21,29)(13,23,24,28)(14,26,27,30)(18,32,34,33)(19,35,36,25)(22,38,40,39)(31,37,41,42)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483563);