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) = C22×C42 |
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 | |
841 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)(17,18)(19,22)(20,21)(23,24);(1,2,4,3)(5,6,8,7)(10,13)(14,16)(18,21)(22,23);(2,3)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24);(1,3,4,2)(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^8|b^2=h,d^2=g,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 | #142793, #142851, #144863, … |
| maximal normal subgroup of | #142793, #142851, #144863, … |
| maximal quotient of | #155767, #155862, #156072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10003);