Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C22×C16 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2,3,4);(6,8)Polycyclic presentation:
2^8|a^2=f,b^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | #93323, #93324, #93356, … |
| maximal normal subgroup of | #93323, #93324, #93356, … |
| maximal quotient of | #134218, #134246, #134272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5771);