Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24);(6,8)(10,11)(14,15)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,d^2=eh,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #209422, #209426, #209438, … |
| maximal normal subgroup of | #209422, #209426, #209438, … |
| maximal quotient of | #153916, #153924, #153928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7296);