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 | |
565 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)(9,10,13,16,14,17,22,23)(11,15,19,12,18,21,24,20);(2,4)(5,6,8,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24);(5,7,8,6)(11,18)(12,20)(15,21)(19,24);(9,12,13,21,14,20,22,15)(10,11,16,19,17,18,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,b^a=be,d^a=dh,c^b=ch,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #128567, #128569, #128580, … |
| maximal normal subgroup of | #128567, #128569, #128580, … |
| maximal quotient of | #183842, #183844, #183850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13483);