Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16260 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21);(6,8)(10,14)(11,16)(17,18,20,22)(19,21,24,23);(5,6)(7,8)(9,11,13,16)(10,12,14,15)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(9,12)(10,11)(13,15)(14,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=ef,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16351, #610837, #610854, … |
| maximal normal subgroup of | C2×#16351, #610837, #610854, … |
| maximal quotient of | #155629, #155637, #155809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9339);