Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
561 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)(9,10,14,17,13,16,22,24)(11,18,20,21,19,15,23,12);(2,4)(5,6,8,7)(9,11)(10,15)(12,17)(13,19)(14,20)(16,18)(21,24)(22,23);(5,7,8,6)(11,19)(12,21)(15,18)(20,23);(9,12,14,18,13,21,22,15)(10,11,17,20,16,19,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,b^a=be,c^b=ch,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #101695, #101697, #101699, … |
| maximal normal subgroup of | #101695, #101697, #101699, … |
| maximal quotient of | #183862, #183937, #183938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13480);