Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,14,10,15,9,11)(7,12,16,13,17,20,18,19);(6,11)(7,16)(8,9)(13,19)(14,15)(17,18);(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,14)(15,20);(2,4)(5,7)(6,13)(8,18)(9,16)(10,17)(11,20)(12,14)(15,19)Polycyclic presentation:
2^8|a^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632114, #6668408, #6668409, … |
| maximal normal subgroup of | #6632114, #6668408, #6668409, … |
| maximal quotient of | #332983, #332989, #332995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23478);