Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,16,19)(11,20,23,13)(12,14,24,15)(17,18,21,22);(2,5)(6,7)(9,11,14,22,16,23,15,18)(10,17,12,20,19,21,24,13);(9,12,14,19,16,24,15,10)(11,21,22,13,23,17,18,20);(1,3,4,8)(2,6,5,7)(9,13,15,21,16,20,14,17)(10,18,24,23,19,22,12,11)Polycyclic presentation:
2^8|a^2=ef,b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #504176, #504798, #510293, … |
| maximal normal subgroup of | #504176, #504798, #510293, … |
| maximal quotient of | #308987, #308992, #308996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19492);