Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 4096 |
| 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 | |
166 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 63 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)(3,4)(5,6,10,17,12,18,11,27)(7,19,22,30,24,35,23,33)(8,16,25,29,26,34,9,15)(13,28,31,36,32,21,14,20);(5,7,8,20,10,22,25,28,12,24,26,36,11,23,9,21)(6,13,29,35,27,14,16,30,18,32,15,19,17,31,34,33);(2,4)(5,7,8,20,10,22,25,28,12,24,26,36,11,23,9,21)(6,14,15,33,27,32,34,35,18,31,29,30,17,13,16,19)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13151, #494919, #494920, … |
| maximal normal subgroup of | C2×#13151, #494919, #494920, … |
| maximal quotient of | #140174, #140175, #140201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6139);