Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
571 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)(3,4)(6,8)(9,10,12,14)(11,13,16,15)(19,22)(20,24);(1,2,4,3)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,22,24);(2,3)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,20,23,24);(1,3,4,2)(9,11,12,16)(10,13,14,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=eg,b^2=h,c^2=ef,d^2=e,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 | #294417, #294418, #294449, … |
| maximal normal subgroup of | #294417, #294418, #294449, … |
| maximal quotient of | #155687, #155815, #155837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9693);