Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups; only the 563 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 563 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 — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(1,3,4,8)(2,5,6,7);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(11,19)(15,23)(18,24)(20,21);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111362, #111365, #111439, … |
| maximal normal subgroup of | #111362, #111365, #111439, … |
| maximal quotient of | #6371127, #6371870, #6372206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53651);