Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11960 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1080 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);(3,4,5,8,6,7,9,10)(11,12,16,22,17,20,32,38,18,23,30,40,14,29,15,21)(13,24,27,37,25,39,26,34,28,41,31,19,33,35,42,36);(4,7)(5,9)(11,13)(12,19)(14,25)(15,31)(16,26)(17,33)(18,28)(20,34)(21,39)(22,24)(23,37)(27,32)(29,36)(30,42)(35,38)(40,41);(4,7)(8,10)(11,14,18,17)(12,20,23,29)(13,25,28,33)(15,30,32,16)(19,34,37,36)(21,22,38,40)(24,35,41,39)(26,31,42,27)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390302);