Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 189 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)(3,15,20,50)(4,21,27,57)(5,12,30,47)(6,11,31,46)(7,32,29,13)(9,34,38,17)(10,39,44,24)(16,51,55,35)(18,37,28,48)(19,36,33,45)(22,58,52,63)(23,43,25,61)(26,41,60,42)(40,53,49,62)(54,56,59,64);(1,3)(2,9)(4,22)(5,28)(6,17)(7,33)(8,20)(10,40)(11,45)(12,15)(13,48)(14,38)(16,25)(18,30)(19,29)(21,56)(23,55)(24,54)(26,62)(27,52)(31,34)(32,37)(35,42)(36,46)(39,59)(41,51)(43,58)(44,49)(47,50)(53,60)(57,64)(61,63);(1,4,5,23,29,60,31,39,8,27,30,25,7,26,6,24)(2,10,11,41,32,61,47,57,14,44,46,42,13,43,12,21)(3,16,17,52,33,59,18,53,20,55,34,22,19,54,28,62)(9,35,15,49,48,64,36,63,38,51,50,40,37,56,45,58)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13443, #499179, #499515, … |
| maximal normal subgroup of | C2×#13443, #499179, #499515, … |
| maximal quotient of | #145361, #145362, #145364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6431);