Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,7,8)(13,14,16,18)(15,19,20,17);(5,6)(7,8)(10,12)(15,20)(17,19);(6,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,8)(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^8|a^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17086, #844862, #844869, … |
| maximal normal subgroup of | C2×#17086, #844862, #844869, … |
| maximal quotient of | #156466, #156468, #156472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10074);