Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,8,14)(3,9,18,34)(4,19,21,53)(5,25,29,41)(6,13,7,12)(10,35,37,22)(11,27,42,28)(15,44,45,64)(16,47,50,63)(17,43,30,33)(20,55,51,52)(23,54,24,40)(26,39,57,38)(31,59,60,46)(32,58,61,49)(36,56,62,48);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,36)(11,32)(12,43)(13,33)(14,34)(15,23)(19,52)(21,51)(22,56)(24,45)(25,47)(26,46)(27,58)(28,49)(29,50)(31,38)(35,48)(37,62)(39,60)(40,44)(41,63)(42,61)(53,55)(54,64)(57,59);(1,4,7,23,8,21,6,24)(2,10,13,38,14,37,12,39)(3,15,17,51,18,45,30,20)(5,26,28,35,29,57,27,22)(9,31,33,62,34,60,43,36)(11,40,25,53,42,54,41,19)(16,48,49,59,50,56,58,46)(32,55,47,64,61,52,63,44)Polycyclic presentation:
2^8|a^2=h,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,f^b=fh,d^c=dh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13397, #499050, #499270, … |
| maximal normal subgroup of | C2×#13397, #499050, #499270, … |
| maximal quotient of | #144717, #144753, #145025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6385);