Finite Group DatabasebetaFinite Group Databasebeta| order | 1472 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C23 · Schur multiplier (H2) = C25 |
| direct product | C23×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C23 | C24 |
| commutator | C2 | C23×C4×C23 |
| Frattini, cosocle | C4 | C24×C23 |
| socle | C2×C23 | C23×C4 |
| prime radical | C2×#41×C23 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C23 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 368 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,27,31,28,30,34,37)(26,29,33,36,32,35,38,39);(24,26)(25,29)(27,33)(28,32)(30,35)(31,36)(34,38)(37,39);(26,32)(29,35)(33,38)(36,39);(25,30)(29,35)(31,37)(36,39)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1472, 249);