Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,17)(8,19)(9,12)(10,16)(11,15)(13,14)(18,20);(6,12,16,15)(7,18,14,19)(8,20)(13,17);(2,4)(5,7)(6,13)(8,12)(9,18)(10,14)(11,19)(15,20)(16,17);(5,8,9,13,10,20,11,17)(6,14,15,19,16,7,12,18)Polycyclic presentation:
2^8|b^2=eg,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151276, #151277, #151436, … |
| maximal normal subgroup of | #151276, #151277, #151436, … |
| maximal quotient of | #317904, #317905, #317906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21236);