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 | #41×C32 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C32 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C23×C16 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 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);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419984);