Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C7 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(5,8)(9,10,11,12,13,14,15);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,7)(6,8)112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 865);