Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1434 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2,5,10,7,12,6,11)(3,13,16,8,17,23,15,24)(4,18,19,25,21,26,20,9)(14,28,30,27,31,22,29,32);(1,3,6,16,7,17,5,15)(2,8,10,13,12,24,11,23)(4,14,20,30,21,31,19,29)(9,22,18,27,25,28,26,32);(1,4,5,19,7,21,6,20)(2,9,10,18,12,25,11,26)(3,14,15,29,17,31,16,30)(8,22,13,27,24,28,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129943, #129944, #138080, … |
| maximal normal subgroup of | #129943, #129944, #138080, … |
| maximal quotient of | #108432, #108433, #108434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3800);