Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| 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 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 933 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:
(1,2)(3,4)(5,6,9,13,10,12,17,7)(8,11,16,19,15,18,20,14);(5,7,9,12,10,13,17,6)(8,14,16,18,15,19,20,11);(2,4)(5,8)(6,11)(7,14)(9,16)(10,15)(12,18)(13,19)(17,20);(8,15)(11,18)(14,19)(16,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147504, #147513, #148023, … |
| maximal normal subgroup of | #147504, #147513, #148023, … |
| maximal quotient of | #195927, #195932, #195934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13978);