Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ 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:
(2,3)(5,6,11,16,9,15,12,27)(7,17,20,34,19,13,10,25)(8,21,24,33,22,36,28,29)(14,26,31,18,30,35,32,23);(1,2,4,3)(5,7,12,10,9,19,11,20)(6,13,27,34,15,17,16,25)(8,18,28,26,22,23,24,35)(14,29,32,36,30,33,31,21);(1,3,4,2)(5,8,9,22)(6,14,15,30)(7,18,19,23)(10,26,20,35)(11,24,12,28)(13,29,17,33)(16,31,27,32)(21,34,36,25)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^c=eg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13134, #494865, #494866, … |
| maximal normal subgroup of | C2×#13134, #494865, #494866, … |
| maximal quotient of | #140106, #140107, #140108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6122);