Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23);(2,4)(6,8)(10,11)(14,15)(17,19,22,21)(18,23,20,24);(1,2,3,4)(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24);(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^8|b^2=eh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#15072, #608185, #608197, … |
| maximal normal subgroup of | C2×#15072, #608185, #608197, … |
| maximal quotient of | #154157, #154249, #154250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8060);