Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
173 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 173 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,4,5,15)(3,10,13,11,14,16,12,8)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(1,3,5,12,7,14,6,13)(2,8,9,10,4,11,15,16)(18,22)(23,31)(24,29)(25,32);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25)Polycyclic presentation:
2^8|a^2=f,b^2=dh,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #131760, #131761, #131772, … |
| maximal normal subgroup of | #131760, #131761, #131772, … |
| maximal quotient of | #98893, #98903, #98911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1675);