Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
282 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3,5,13,6,14,4,12)(2,7,8,15,10,16,9,11)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fgh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #105437, #105438, #105439, … |
| maximal normal subgroup of | #105437, #105438, #105439, … |
| maximal quotient of | #95223, #95224, #95225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1064);