Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#555 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
401 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,6,7,5,3,4,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,3,5,2)(4,8,6,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,4,5,6)(2,7,3,8)(10,15)(12,21)(14,17)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=h,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | #132182, #132183, #132185, … |
| maximal normal subgroup of | #132182, #132183, #132185, … |
| maximal quotient of | #97417, #97424, #97428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1385);