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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1982 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 236 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:
(1,2,6,13,7,12,22,31,8,14,21,10,5,23,4,11)(3,15,19,29,18,30,17,26,20,25,16,9,24,27,32,28);(1,3,7,24,8,20,5,18)(2,9,12,26,14,29,23,28)(4,16,6,17,22,19,21,32)(10,25,11,30,13,15,31,27);(1,4,5,21,8,22,7,6)(2,10,12,11,14,13,23,31)(3,16,18,32,20,19,24,17)(9,25,26,30,29,15,28,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136867, #136870, #136873, … |
| maximal normal subgroup of | #136867, #136870, #136873, … |
| maximal quotient of | #140848, #140849, #140850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6247);