Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(8,9)(11,12)(13,18)(14,24)(15,27)(16,31)(17,21)(19,25)(20,29)(22,38)(23,37)(26,41)(28,33)(30,40)(32,36)(34,42)(35,39);(3,4,6,7)(5,9,10,8)(11,13,17,16)(12,18,21,31)(14,22,26,32)(15,23,30,29)(19,33,35,34)(20,27,37,40)(24,38,41,36)(25,28,39,42);(11,14,17,26)(12,19,21,35)(13,22,16,32)(15,28,30,42)(18,33,31,34)(20,36,37,38)(23,39,29,25)(24,40,41,27);(3,5,6,10)(4,8,7,9)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531056);