Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10,13,19,14,17,12,18)(11,20,21,24,22,15,23,16);(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16,17,24)(12,21)(13,23)(14,22)(15,18,20,19)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eh,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | #130530, #130531, #130532, … |
| maximal normal subgroup of | #130530, #130531, #130532, … |
| maximal quotient of | #99164, #99167, #99173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1764);