Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 483 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,12,10,14,8,13)(7,15,18,20,16,19,17,11);(5,7,8,17,10,16,9,18)(6,11,12,15,14,20,13,19);(2,4)(6,12)(7,16)(8,9)(11,15)(13,14)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11142, #357839, C4×#2048, … |
| maximal normal subgroup of | C2×#11142, #357839, C4×#2048, … |
| maximal quotient of | #116995, #117037, #117041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4130);