Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25 |
| direct product | 3−1+2×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2×C3 | C24×C32 |
| prime radical | C32×2−1+4 | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(11,15)(13,21)(16,18)(19,25);(2,5,6)(4,8,9)(10,11,14,15)(12,18,20,16)(13,17,21,22)(19,24,25,23);(10,12,14,20)(11,16,15,18)(13,19,21,25)(17,23,22,24);(10,13)(11,17)(12,19)(14,21)(15,22)(16,23)(18,24)(20,25)187 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2665);