Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C22×C42 | C22 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
201 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,19,20,23,22,24);(2,5,7,6)(9,11)(10,13,14,15)(12,16)(17,19,22,23,20,24,21,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7045, #96127, #96128, … |
| maximal normal subgroup of | C2×#7045, #96127, #96128, … |
| maximal quotient of | #92811, #92812, #92813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33);