Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 239 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,11,13,10,7,9,14)(8,12,16,20,18,15,17,19);(1,2,3,4)(5,7,10,6)(8,15,18,12)(9,14,11,13)(16,20,17,19);(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,19)(14,20);(5,9,10,11)(6,13,7,14)(8,16,18,17)(12,19,15,20)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147804, #147805, #147837, … |
| maximal normal subgroup of | #147804, #147805, #147837, … |
| maximal quotient of | #193792, #193801, #193805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13944);