Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,22,20,23,19,24);(1,2,4,5)(3,8,7,6)(9,11)(10,13)(12,15)(14,16)(17,19,20,21)(18,22,23,24);(2,5)(6,8)(10,14)(11,15);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19,20,21)(18,22,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=deh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23341, #6407535, #6407537, … |
| maximal normal subgroup of | C2×#23341, #6407535, #6407537, … |
| maximal quotient of | #357296, #357321, #357322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16329);