Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
981 conjugacy classes of subgroups.
| 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)(12,13)(15,16);(2,5)(6,8)(17,18,20,22)(19,23,24,21);(10,13)(14,16)(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,14,13,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34552⋊C3 |
| maximal subgroup of | #121996, #129957, #237996, … |
| maximal normal subgroup of | #121996, #129957, #237996, … |
| maximal quotient of | #514017, #514026, C4×#2396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27540);