Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C4×C16 |
| Frattini, cosocle | C23×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 157 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,11,16,20,15,19,13,18,23,22,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,19,24,17,15,22,12,20,23,10,16,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=g,g^2=h,b^a=bc,c^a=cf,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7199, #127050, #127051, … |
| maximal normal subgroup of | C2×#7199, #127050, #127051, … |
| maximal quotient of | #92995, #92996, #92997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 187);