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 | C2×#508 | 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:
(1,2,7,13,8,14,30,46,9,15,6,31,29,45,5,12)(3,16,20,39,21,44,18,47,22,50,19,10,34,37,32,38)(4,23,25,48,26,59,24,11,27,41,28,42,35,40,33,43)(17,51,53,58,54,36,56,49,55,62,60,64,57,63,52,61);(1,3,8,34,9,22,29,21)(2,10,14,47,15,39,45,38)(4,17,35,54,27,55,26,57)(5,19,7,18,30,20,6,32)(11,36,48,62,43,63,42,51)(12,44,13,16,46,37,31,50)(23,58,40,49,41,64,59,61)(24,53,25,56,33,60,28,52);(1,4)(2,11)(3,17)(5,28)(6,33)(7,24)(8,35)(9,27)(10,36)(12,23)(13,40)(14,48)(15,43)(16,49)(18,53)(19,52)(20,56)(21,57)(22,55)(25,30)(26,29)(31,59)(32,60)(34,54)(37,64)(38,51)(39,63)(41,46)(42,45)(44,58)(47,62)(50,61)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,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13016, #494512, #495719, … |
| maximal normal subgroup of | C2×#13016, #494512, #495719, … |
| maximal quotient of | #136689, #136690, #136691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6004);