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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1947 |
| prime radical | C23 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,14,17,30,18,24,16,28)(4,11,5,21,20,29,19,10)(9,22,25,32,27,31,26,15);(1,3,7,17,8,18,6,16)(2,9,12,25,13,27,23,26)(4,15,5,22,20,32,19,31)(10,24,11,28,21,14,29,30);(1,4,6,19,8,20,7,5)(2,10,12,11,13,21,23,29)(3,15,16,31,18,32,17,22)(9,24,25,28,27,14,26,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136326, #136328, #136389, … |
| maximal normal subgroup of | #136326, #136328, #136389, … |
| maximal quotient of | #140380, #140381, #140401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6170);