Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
261 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,5,7,8)(4,6)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24);(1,3)(2,5,6,8)(4,7)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(2,6)(5,8)(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138916, #138917, #138924, … |
| maximal normal subgroup of | #138916, #138917, #138924, … |
| maximal quotient of | #102061, #102064, #102067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2770);