Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 32768 |
| 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 | |
1099 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)(7,14)(8,12)(9,17)(10,13)(11,15)(16,20)(18,19);(5,7)(6,11)(8,15)(9,18)(10,16)(12,20)(13,19)(14,17);(2,4)(8,9)(11,19)(12,17)(15,18);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,20,19,14)Polycyclic presentation:
2^8|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #376207, #381472, #388379, … |
| maximal normal subgroup of | #376207, #381472, #388379, … |
| maximal quotient of | #233670, #233674, #233678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16390);