Finite Group DatabasebetaFinite Group Databasebeta| order | 624 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C13 · Schur multiplier (H2) = C1 |
| direct product | C3×C13×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C13 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C13 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C13 ◁ C3×C4×C13 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 624 |
A minimal faithful permutation representation — 2 generators on 32 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)(25,32,31,30,29,28,27,26);(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)273 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(624, 116);