Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | Aut(G) order 8192 |
| 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 | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
343 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2,5,8,6,3,4,7)(10,14)(12,19)(15,23)(16,20)(17,21)(22,24);(1,3,6,2)(4,7,5,8)(9,10,13,16)(11,15,18,24)(12,20,19,14)(17,22,21,23);(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | #139540, #139541, #139542, … |
| maximal normal subgroup of | #139540, #139541, #139542, … |
| maximal quotient of | #96941, #96948, #97131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1344);