Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 125 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)(3,4)(9,10)(11,18)(12,14)(13,15)(16,23)(17,21)(19,22)(20,24);(5,6)(7,8)(10,14)(11,19)(12,21)(15,17)(18,24)(20,23);(2,4)(6,8)(10,15)(14,17)(16,22)(18,24);(2,4)(9,11,12,19,13,20,21,23)(10,16,17,24,15,22,14,18)Polycyclic presentation:
2^8|d^2=e,e^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cg,d^b=de,e^b=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123212, #123228, #6632656, … |
| maximal normal subgroup of | #123212, #123228, #6632656, … |
| maximal quotient of | #339705, #339706, #339708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23833);