Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator, prime radical | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
442 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 187 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 — 4 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14);(17,19,22,28)(18,23,26,31)(20,29,21,27)(24,32,25,30);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=gh,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dgh55 conjugacy classes — view the full character table.
| Fitting subgroup of | #17430⋊C3 |
| maximal subgroup of | #138781, #138783, #138829, … |
| maximal normal subgroup of | #138781, #138783, #138829, … |
| maximal quotient of | #164353, #164356, #164432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10418);