Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
699 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 263 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,11)(12,15,16,14)(18,19)(22,23);(10,11)(12,16)(17,18,21,19)(20,23,24,22);(1,2,4,3)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22);(1,3,4,2)(9,12,13,16)(10,14,11,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=eg,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #99383, #99384, #99385, … |
| maximal normal subgroup of | #99383, #99384, #99385, … |
| maximal quotient of | #153966, #153981, #154011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7455);