Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10,12,18,14,17,13,16)(11,15,19,23,21,22,20,24);(1,2,4,3)(9,11,12,19,14,21,13,20)(10,15,18,23,17,22,16,24);(6,8)(10,16)(11,19)(12,13)(17,18)(20,21)(23,24);(1,3,4,2)(10,17)(15,22)(16,18)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=egh,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28468, #6465619, #6465629, … |
| maximal normal subgroup of | C2×#28468, #6465619, #6465629, … |
| maximal quotient of | #321304, #321311, #321317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21456);