Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1879 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,7,14,8,15,31,46,9,16,6,13,30,45,5,12)(3,17,20,39,21,44,19,49,22,47,29,10,34,37,32,38)(4,23,26,50,27,48,24,11,28,41,25,42,35,40,33,43)(18,52,54,59,55,36,60,61,56,62,57,64,58,63,53,51);(1,3,8,34,9,22,30,21)(2,10,15,49,16,39,45,38)(4,18,35,55,28,56,27,58)(5,29,7,19,31,20,6,32)(11,36,50,62,43,63,42,52)(12,44,14,17,46,37,13,47)(23,59,40,61,41,64,48,51)(24,54,26,60,33,57,25,53);(1,4)(2,11)(3,18)(5,26)(6,24)(7,33)(8,35)(9,28)(10,36)(12,41)(13,40)(14,48)(15,50)(16,43)(17,51)(19,57)(20,53)(21,58)(22,56)(23,46)(25,31)(27,30)(29,60)(32,54)(34,55)(37,59)(38,52)(39,63)(42,45)(44,64)(47,61)(49,62)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13021, #494523, #495719, … |
| maximal normal subgroup of | C2×#13021, #494523, #495719, … |
| maximal quotient of | #136693, #136694, #136727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6009);