Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×C8×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(11,12,13,15,14,16,17,18);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10);(3,5,6,9)(4,8,7,10)160 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C8×#41⋊C3 |
| maximal subgroup of | #105564, #105566, #105636, … |
| maximal normal subgroup of | #105564, #105566, #105636, … |
| maximal quotient of | C2×C8×#117, C2×C4×C8×D4, C2×C4×C8×Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53184);