Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
775 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 775 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,14)(9,13)(10,12)(11,16)(17,20)(18,19);(5,6,10,14,9,13,8,12)(7,11,18,20,17,19,16,15);(1,2,3,4)(5,6,9,13)(7,11,17,19)(8,12,10,14)(15,18,20,16);(5,7,10,18,9,17,8,16)(6,11,14,20,13,19,12,15)Polycyclic presentation:
2^8|b^2=eh,c^2=f,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 | #501028, #501032, #501034, … |
| maximal normal subgroup of | #501028, #501032, #501034, … |
| maximal quotient of | #299256, #299259, #299263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18793);