Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 3 generators on 24 points:
(2,5,6,7)(9,10,13,18,14,17,19,22)(11,15,21,23,20,24,12,16);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15,17,24)(12,19)(13,21)(14,20)(16,22,23,18);(9,12,14,21)(10,16,17,23)(11,19,20,13)(15,22,24,18)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=h,b^a=bd,d^a=df,f^a=fh,d^b=dg,e^b=efgh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128792, #128793, #128820, … |
| maximal normal subgroup of | #128792, #128793, #128820, … |
| maximal quotient of | #124089, #124121, #124191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4709);