Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C23×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 51 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)(9,10,12,18,15,11,16,20,14,17,13,23,22,19,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,15,17,24,23,14,19,12,21,22,10,16,18)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,e^a=eh,c^b=ch,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7201, #127058, #127059, … |
| maximal normal subgroup of | C2×#7201, #127058, #127059, … |
| maximal quotient of | #93003, #93004, #93005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 189);