Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
725 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 725 conjugacy classes of subgroups — too many to draw. Only its 137 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)(5,6,10,12,8,13,9,14)(7,15,18,19,17,11,16,20);(2,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,20)(13,15)(14,19);(6,12)(7,16)(9,10)(13,14)(17,18)(19,20);(2,4)(6,13)(11,15)(12,14)(19,20)Polycyclic presentation:
2^8|a^2=fh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #267987, #267993, #268003, … |
| maximal normal subgroup of | #267987, #267993, #268003, … |
| maximal quotient of | #267052, #267057, #267064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17387);