Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,17,14,18,23,20,15,19,24,11,12,16,22,21);(1,2,4,7,5,8,3,6)(9,11,13,18,14,21,23,19,15,17,24,16,12,20,22,10)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=efh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7168, #126761, #126798, … |
| maximal normal subgroup of | C2×#7168, #126761, #126798, … |
| maximal quotient of | #92935, #92936, #92937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 156);