Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,4,6,5,7,3,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,e^b=ef,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130475, #130477, #130479, … |
| maximal normal subgroup of | #130475, #130477, #130479, … |
| maximal quotient of | #124747, #124748, #124762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4514);