Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#12887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
678 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);(3,4)(5,8)(6,7)(9,10)(11,12)(13,23,27,38)(14,29,28,42)(15,31)(16,32)(17,22)(18,26,36,39)(19,30,37,40)(20,25)(21,33)(24,35)(34,41);(4,7)(8,10)(11,13,24,39,16,28,41,30,17,27,15,26,33,14,25,40)(12,18,34,38,21,19,35,42,22,36,20,23,32,37,31,29);(3,5,6,9)(4,8,7,10)(11,14,15,30,16,13,25,26,17,28,24,40,33,27,41,39)(12,19,20,29,21,36,31,38,22,37,34,42,32,18,35,23)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401274);