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, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1041 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,7)(4,5)(6,8)(9,11,12,16)(10,13,15,14)(18,21)(22,23);(2,5)(6,7)(9,10,12,15)(11,14,16,13)(17,18)(19,23)(20,21)(22,24);(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,19,20,24)(18,22,21,23);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^8|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 | #36573⋊C3 |
| maximal subgroup of | #130957, #224209, #224225, … |
| maximal normal subgroup of | #130957, #224209, #224225, … |
| maximal quotient of | #519251, #519426, #519617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29561);