Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C22×C4⋊C4 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
187 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 2 generators on 24 points:
(2,5,6,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(1,2)(3,5,7,8)(4,6)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,e^2=f,b^a=bc,c^a=cf,e^a=egh,c^b=cg,d^b=dfh,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #93217, #93218, #93219, … |
| maximal normal subgroup of | #93217, #93218, #93219, … |
| maximal quotient of | #93466, #93497, #93557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 314);