Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,16,21)(13,17,22,30)(14,19,25,33)(15,18,27,20)(23,29,35,40)(24,31,36,28)(26,32,38,34)(37,41,42,39);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,23)(15,24)(16,22)(18,31)(19,29)(20,28)(21,30)(25,35)(26,37)(27,36)(32,41)(33,40)(34,39)(38,42);(5,10)(7,9)(12,18)(13,22)(15,27)(17,28)(19,32)(20,21)(23,35)(26,38)(29,39)(30,31)(33,34)(40,41);(4,8)(7,9)(11,14)(12,19)(13,23)(15,26)(16,25)(17,29)(18,32)(20,34)(21,33)(22,35)(24,37)(27,38)(28,39)(30,40)(31,41)(36,42);(14,25)(19,33)(23,35)(26,38)(29,40)(32,34)(37,42)(39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482729);