Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,20)(9,25)(10,27)(11,16)(12,18)(13,22)(14,26)(17,31)(19,33)(21,35)(23,32)(24,36)(28,30)(29,34);(1,3,4,2)(5,7,9,18,11,19,26,10)(6,12,14,15,16,27,25,33)(8,17,21,34,24,22,23,28)(13,20,29,32,31,36,30,35);(5,8,9,21,11,24,26,23)(6,13,14,29,16,31,25,30)(7,17,18,34,19,22,10,28)(12,20,15,32,27,36,33,35)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^c=egh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13135, #494870, #494871, … |
| maximal normal subgroup of | C2×#13135, #494870, #494871, … |
| maximal quotient of | #140110, #140138, #140215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6123);