Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C27 · Schur multiplier (H2) = C2 |
| direct product | C27×#125 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C27 | C23 |
| commutator | C22 | C23×C27 |
| Frattini, cosocle | C22×C9 | C23×C3 |
| socle, prime radical | C22×C3 | C23×C9 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C27 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(30,34)(32,35)(36,37,39,41)(38,40,42,43);(28,29,31,33)(30,32,34,35)(36,37)(38,40)(39,41)(42,43);(28,30)(29,32)(31,34)(33,35)(36,38,39,42)(37,40,41,43)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 181);