Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3830 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);(3,4,5,8,6,7,9,10)(11,12)(13,21,24,17)(14,18)(15,19)(16,20)(22,35,38,29)(23,36,40,30)(25,31)(26,32)(27,33)(28,34)(37,42,39,41);(11,13)(12,17)(14,22)(15,23)(16,24)(18,29)(19,30)(20,21)(25,37)(26,38)(27,39)(28,40)(31,41)(32,35)(33,42)(34,36);(4,7)(8,10)(11,14,16,26)(12,18,20,32)(13,22,24,38)(15,27,28,25)(17,29,21,35)(19,33,34,31)(23,39,40,37)(30,42,36,41);(11,15,16,28)(12,19,20,34)(13,23,24,40)(14,25,26,27)(17,30,21,36)(18,31,32,33)(22,37,38,39)(29,41,35,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249762);