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) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C2×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,8)(15,19)(18,20);(1,2,4,3)(9,10,11,12)(14,17)(18,20);(2,3)(5,6,7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(1,3,4,2)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|b^2=egh,c^2=f,d^2=e,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 | #258055, #258061, #258067, … |
| maximal normal subgroup of | #258055, #258061, #258067, … |
| maximal quotient of | #155717, #155843, #155845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9807);