Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3515 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2260 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);(4,8,9,10)(11,12)(13,21,24,17)(14,25,27,34)(15,20)(16,19)(18,26,32,28)(22,37)(23,36,41,31)(29,33)(30,38)(35,40)(39,42);(3,4)(5,8)(6,10)(7,9)(11,13)(12,17)(14,22)(15,23)(16,24)(18,30)(19,21)(20,31)(25,42)(26,38)(27,39)(28,40)(29,41)(32,35)(33,36)(34,37);(3,5,7,6)(4,8,9,10)(11,14,16,27)(12,18,19,32)(13,22,24,39)(15,28,29,26)(17,30,21,35)(20,34,33,25)(23,40,41,38)(31,37,36,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93316);