Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C32 · Schur multiplier (H2) = C22×C4 |
| direct product | C4⋊C4×C32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C22 |
| commutator | C2 | C2×C4×C32 |
| Frattini, cosocle | C22×C16 | C23 |
| socle, prime radical | C23 | C22×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40);(2,4)(5,6,7,8);(1,2,3,4)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59179);