Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6,8,7);(2,4)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(5,7,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,g^2=h,b^a=be,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #100524, #100528, #100532, … |
| maximal normal subgroup of | #100524, #100528, #100532, … |
| maximal quotient of | #183330, #183331, #183332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13418);