Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(10,13)(14,16)(17,18,20,22)(19,23,24,21);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8)(17,18)(19,21)(20,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#59252, #7665767, #7666086, … |
| maximal normal subgroup of | C2×#59252, #7665767, #7666086, … |
| maximal quotient of | #531657, #531728, #531957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52240);