Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
813 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 813 conjugacy classes of subgroups — too many to draw. Only its 447 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) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,5,8)(10,14)(15,16)(18,22)(23,24);(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,20,24);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,23,22,24);(1,4,6,2)(3,7,5,8)(9,11,13,12)(10,15,14,16);(1,5,6,3)(2,8,4,7)(9,12,13,11)(10,16,14,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=fgh,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #36608⋊C3 |
| maximal subgroup of | #130677, #224658, #224674, … |
| maximal normal subgroup of | #130677, #224658, #224674, … |
| maximal quotient of | #519252, #519427, #519619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29596);