Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7329 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C16 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle, prime radical | C23 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(11,12,13,18,14,19,21,30,15,20,22,31,24,32,33,40,16,17,23,27,25,28,34,37,26,29,35,38,36,39,41,42);(3,4,5,7,6,8,9,10)(12,17)(18,27)(19,28)(20,29)(30,37)(31,38)(32,39)(40,42)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56687);