Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| 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 | |
447 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 447 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)(5,6,9,12)(7,14,17,20)(8,13,15,11)(10,18,19,16);(5,6,8,11,9,12,15,13)(7,10,16,14,17,19,18,20);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20);(5,7,15,18,9,17,8,16)(6,10,13,20,12,19,11,14)Polycyclic presentation:
2^8|a^2=f,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,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501236, #501237, #501240, … |
| maximal normal subgroup of | #501236, #501237, #501240, … |
| maximal quotient of | #277277, #277279, #277284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17785);