Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 3 generators on 20 points:
(2,3)(5,6,9,12,10,14,8,13)(7,11,15,19,16,20,17,18);(1,2,4,3)(5,7)(6,11)(8,17)(9,15)(10,16)(12,19)(13,18)(14,20);(1,3,4,2)(6,12)(8,9)(11,18)(13,14)(15,17)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131417, #131418, #131419, … |
| maximal normal subgroup of | #131417, #131418, #131419, … |
| maximal quotient of | #120009, #120010, #120041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4213);