Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,19)(10,15)(11,16)(12,18)(14,20);(5,7,10,18,11,8,20,19)(6,12,14,17,16,9,15,13);(2,4)(5,8)(6,13)(7,11)(9,15)(10,19)(12,14)(16,17)(18,20);(5,9,10,13,11,12,20,17)(6,8,14,19,16,7,15,18)Polycyclic presentation:
2^8|b^2=e,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24292, #6442709, #6442710, … |
| maximal normal subgroup of | C2×#24292, #6442709, #6442710, … |
| maximal quotient of | #263075, #263078, #263081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17280);