Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 576 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C16×C27 |
| multiplicative | (ℤ/1299)ˣ, (ℤ/1377)ˣ, (ℤ/1732)ˣ, (ℤ/2598)ˣ, (ℤ/2754)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8×C9 | C22×C3 |
| socle, prime radical | C22×C3 | C8×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(17,18)864 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 59);