Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
213 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 213 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,13,16,12,15,19,21)(11,14,18,23,17,22,20,24);(1,2,4,7,6,8,5,3)(9,11,13,20,12,17,19,18)(10,14,21,23,15,22,16,24);(1,3,5,8,6,7,4,2)(10,15)(14,22)(16,21)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eh,d^2=gh,e^2=gh,f^2=h,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 | C2×#9797, #205867, #205872, … |
| maximal normal subgroup of | C2×#9797, #205867, #205872, … |
| maximal quotient of | #102063, #102079, #102174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2785);