Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C16×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C4⋊C4 |
| prime radical | C23 | C8×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36387);