Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
705 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 705 conjugacy classes of subgroups — too many to draw. Only its 175 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)(11,16)(12,21)(13,23)(14,17)(15,19)(18,22)(20,24);(5,6)(7,8)(9,11,14,20)(10,15,17,21)(12,22,19,13)(16,18,24,23);(2,4)(6,8)(9,12,13,11,14,19,22,20)(10,16,18,21,17,24,23,15);(10,17)(15,21)(16,24)(18,23)Polycyclic presentation:
2^8|b^2=h,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #394692, #394693, #395429, … |
| maximal normal subgroup of | #394692, #394693, #395429, … |
| maximal quotient of | #238906, #238908, #238910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16533);