Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(7,14)(8,16)(9,10)(17,18)(19,20);(1,2)(3,4)(5,6,11,7)(8,13,17,15)(9,14,10,12)(16,19,18,20);(2,4)(5,7,11,6)(8,15,17,13)(9,14,10,12)(16,19,18,20);(5,8)(6,13)(7,15)(9,18)(10,16)(11,17)(12,19)(14,20)Polycyclic presentation:
2^8|b^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^c=fh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23881, #6437472, #6437474, … |
| maximal normal subgroup of | C2×#23881, #6437472, #6437474, … |
| maximal quotient of | #244661, #244662, #244667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16869);