Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
679 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 135 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)(5,6,10,7)(8,16,18,20)(9,14,11,13)(12,19,15,17);(5,7,9,14,10,6,11,13)(8,15,17,20,18,12,19,16);(2,4)(5,6)(7,10)(8,15)(9,13)(11,14)(12,18)(16,19)(17,20);(5,8,11,19,10,18,9,17)(6,12,14,20,7,15,13,16)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #499785, #501814, #509105, … |
| maximal normal subgroup of | #499785, #501814, #509105, … |
| maximal quotient of | #263077, #263080, #263083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17282);