Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C27 · Schur multiplier (H2) = C22 |
| direct product | C4×Q8×C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C27 | C22 |
| commutator | C2 | C22×C4×C27 |
| Frattini, cosocle | C22×C9 | C23×C3 |
| socle, prime radical | C22×C3 | C23×C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C27 ◁ G | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4)(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);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)540 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 175);