Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12814 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,6)(7,8,9,10)(11,12)(13,21)(14,17)(15,27)(16,30)(18,23)(19,29)(20,26)(22,37)(24,41)(25,38)(28,36)(31,35)(32,39)(33,40)(34,42);(3,4,5,6)(12,17)(18,32)(19,33)(20,30)(21,37)(27,38)(34,36)(35,41);(11,13,14,22)(12,18,17,32)(15,28,25,42)(16,31,26,24)(19,35,33,41)(20,36,30,34)(21,38,37,27)(23,40,39,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400954);