Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11963 |
| 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 | |
1090 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)(11,12,16,21,14,23,30,20,17,22,33,38,31,19,15,32)(13,24,27,36,29,37,42,40,28,39,41,18,25,35,26,34);(4,7)(5,9)(11,13)(12,18)(14,25)(15,27)(16,26)(17,28)(19,34)(20,37)(21,39)(22,36)(23,40)(24,38)(29,31)(30,41)(32,35)(33,42);(4,7)(8,10)(11,14,17,31)(12,19,22,23)(13,25,28,29)(15,16,30,33)(18,34,36,40)(20,21,32,38)(24,37,39,35)(26,41,42,27)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390311);