Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
340 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2)(3,7,8,5)(4,6)(9,10)(11,13)(12,15)(14,16)(17,18,20,23,21,24,19,22);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15);(17,19,21,20)(18,22,24,23)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,e^2=h,b^a=bd,d^a=df,d^b=dg,e^b=efg76 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#327, #96800, #96802, … |
| maximal normal subgroup of | C8×#327, #96800, #96802, … |
| maximal quotient of | #96229, C4×#1270, C4×#1271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 553);