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)(9,10,14,17,13,16,22,12)(11,18,21,19,20,15,23,24);(2,4)(5,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23);(9,12,22,16,13,17,14,10)(11,19,23,18,20,24,21,15);(10,16)(11,20)(12,17)(21,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,f^2=h,b^a=be,d^a=dh,d^b=dh,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #108444, #108445, #108449, … |
| maximal normal subgroup of | #108444, #108445, #108449, … |
| maximal quotient of | #183329, M16×C4⋊C8, #183533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13429);