Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C2×D4 · Out(G) = C25×D4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C8×#137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8);(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(10,15)(14,23)(16,20)(17,24)88 conjugacy classes — view the full character table.
| maximal subgroup of | #136743, #136744, #137872, … |
| maximal normal subgroup of | #136743, #136744, #137872, … |
| maximal quotient of | C8×#419, C8×#417, C8×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10839);