Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| 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 | |
631 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 247 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:
(10,13)(14,16)(17,18,20,22)(19,21,24,23);(5,6,7,8)(17,18)(19,21)(20,22)(23,24);(1,2,3,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,22)(21,24);(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #116135, #188501, #188504, … |
| maximal normal subgroup of | #116135, #188501, #188504, … |
| maximal quotient of | #153940, #153942, #153949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7344);