Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,8)(5,6)(9,10)(11,12)(13,17,22,33)(14,24)(15,28)(16,19)(18,26)(20,30)(21,37,25,42)(23,40,39,32)(27,36)(29,35)(31,41,34,38);(3,4,5,10,7,9,6,8)(11,13,16,22)(12,17)(14,25,27,21)(15,23,30,39)(18,34)(19,33)(20,32)(24,42)(26,41,29,38)(28,40)(31,35)(36,37);(3,5,7,6)(4,8,9,10)(11,14,16,27)(12,18,19,35)(13,21,22,25)(15,29,30,26)(17,31,33,34)(20,36,28,24)(23,41,39,38)(32,42,40,37)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106372);