Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25 |
| direct product | C2×Q8×C32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C2×C16 | C24 |
| socle, prime radical | C23 | C22×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
362 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);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419978);