Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(1,2)(3,4)(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20);(2,4)(5,6)(7,11)(8,12)(9,18)(10,13)(14,15)(16,20)(17,19);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24802, #6443043, #6443048, … |
| maximal normal subgroup of | C2×#24802, #6443043, #6443048, … |
| maximal quotient of | #277851, #277854, #277857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17790);