Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
329 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 155 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,22)(23,24);(2,6)(5,7)(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22);(10,11)(15,19)(16,17)(22,23)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cgh,d^a=dh,e^a=eg,d^b=dh,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #137822, #137852, #210357, … |
| maximal normal subgroup of | #137822, #137852, #210357, … |
| maximal quotient of | #182477, #182478, #183481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10967);