Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(1,3,6,2)(4,7,5,8)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105800, #105801, #106002, … |
| maximal normal subgroup of | #105800, #105801, #106002, … |
| maximal quotient of | #95350, #95353, #95354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1061);