Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1845 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1845 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,11)(8,9)(12,19)(13,16)(14,17)(18,20);(5,6)(7,12)(8,11)(9,16)(10,13)(14,18)(15,20)(17,19);(5,7)(6,12)(8,14)(9,17)(10,15)(11,18)(13,20)(16,19);(5,8,10,9)(6,11,13,16)(7,14,15,17)(12,18,20,19)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147759, #147760, #147762, … |
| maximal normal subgroup of | #147759, #147760, #147762, … |
| maximal quotient of | #184932, #184938, #184944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13606);