Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,13,18,14,16,12,17)(11,19,22,15,21,24,20,23);(1,2,6,8,4,7,3,5)(9,11)(10,15)(12,22)(13,20)(14,21)(16,23)(17,24)(18,19);(2,7)(5,8)(10,16)(12,13)(15,23)(20,22)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132503, #132504, #132519, … |
| maximal normal subgroup of | #132503, #132504, #132519, … |
| maximal quotient of | #100069, #100085, #100761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2389);