Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
471 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(2,6)(3,4)(7,8)(10,15,17,21)(16,22,24,23);(1,2)(3,6)(4,8)(5,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,20)(13,18)(14,19)(15,22)(17,24)(21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=gh,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #221185, #221187, #221192, … |
| maximal normal subgroup of | #221185, #221187, #221192, … |
| maximal quotient of | #160646, #160647, #160652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10616);