Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C26×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C27 | C22 |
| commutator, Frattini, cosocle | C2 | C28 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
443586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 20 points:
(1,2);(3,4);(5,6);(7,8);(9,10);(11,12);(13,14,15,16)(17,20,19,18);(13,17,15,19)(14,18,16,20)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494203);