Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13429 |
| automorphism | Aut(G) = #72317013 (order 4096) · Inn(G) = #2044 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2044 |
| commutator, Frattini, prime radical, cosocle | C2×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12)(3,13,17,34)(4,18,22,36)(5,23,26,10)(6,11,28,37)(8,24,31,15)(9,27,33,21)(14,40,19,49)(16,39,45,63)(20,51,53,48)(25,47,54,35)(29,58,60,64)(30,56,61,43)(32,52,55,57)(38,44,62,42)(41,59,50,46);(1,3,7,17)(2,8,12,31)(4,19,22,14)(5,24,26,15)(6,16,28,45)(9,29,33,60)(10,34,23,13)(11,30,37,61)(18,46,36,59)(20,41,53,50)(21,42,27,44)(25,56,54,43)(32,62,55,38)(35,63,47,39)(40,51,49,48)(52,58,57,64);(1,4,5,20,7,22,26,53)(2,9,10,32,12,33,23,55)(3,14,15,41,17,19,24,50)(6,27,25,57,28,21,54,52)(8,29,13,38,31,60,34,62)(11,36,35,51,37,18,47,48)(16,44,43,58,45,42,56,64)(30,46,39,49,61,59,63,40)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,e^c=eh,f^c=fh,g^c=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151238, #151239, #151270, … |
| maximal normal subgroup of | #151238, #151239, #151270, … |
| maximal quotient of | #144673, #144709, #144741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6419);