Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 95 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)(5,6,9,16,12,17,11,26)(7,18,20,28,23,34,22,36)(8,14,24,31,25,32,10,15)(13,27,29,35,30,21,33,19);(2,4)(5,7,10,21,11,22,25,35,12,23,24,27,9,20,8,19)(6,13,14,28,16,29,31,34,17,30,32,36,26,33,15,18);(5,8,9,24,12,25,11,10)(6,14,16,31,17,32,26,15)(7,19,20,27,23,35,22,21)(13,28,29,34,30,36,33,18)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh88 conjugacy classes — view the full character table.
| maximal subgroup of | #128673, #128674, #128725, … |
| maximal normal subgroup of | #128673, #128674, #128725, … |
| maximal quotient of | #128533, #128589, #128601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5629);