Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(11,12,13,15,14,16,17,18);(3,4)(5,6)(7,8,9,10);(4,6)160 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#323, #96672, #96674, … |
| maximal normal subgroup of | C8×#323, #96672, #96674, … |
| maximal quotient of | C2×C8×C22⋊C8, C2×#8148, C2×#8145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13368);