Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,7,10)(5,8,6,9)(11,12,16,20)(13,21,24,40)(14,25,29,39)(15,30,27,19)(17,32,34,22)(18,35,37,28)(23,41,31,36)(26,38,42,33);(4,8)(5,6)(9,10)(11,13,16,24)(12,17,20,34)(14,26,29,42)(15,31,27,23)(18,36,37,41)(19,38,30,33)(21,39,40,25)(22,35,32,28);(4,9)(5,6)(8,10)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396135);