Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4006 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,18)(15,19)(16,20)(21,31)(22,29)(23,30)(24,32)(25,33)(26,34)(27,35)(28,37)(36,41)(38,40)(39,42);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,21)(15,22)(16,23)(18,31)(19,29)(20,30)(24,36)(25,37)(26,38)(27,39)(28,33)(32,41)(34,40)(35,42);(11,14,16,25)(12,18,20,33)(13,21,23,37)(15,26,27,24)(17,28,30,31)(19,34,35,32)(22,38,39,36)(29,41,42,40);(4,8)(7,9)(11,15,16,27)(12,19,20,35)(13,22,23,39)(14,24,25,26)(17,29,30,42)(18,32,33,34)(21,36,37,38)(28,40,31,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359068);