Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C82 |
| Frattini, cosocle | C22×C42 | C22 |
| socle | C23 | #95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
237 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 95 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)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16);(1,3)(2,5)(4,8)(6,7)(17,18,19,21,20,22,23,24)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=h,b^a=bc,c^a=cf,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #92851, #92852, #92875, … |
| maximal normal subgroup of | #92851, #92852, #92875, … |
| maximal quotient of | #92806, #92807, #92814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4);