Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C4×#511819 (order 2048) · Inn(G) = #127 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1955 |
| prime radical | #575 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,10,6,11,21,32,7,12,22,33,23,34,47,58)(3,8,15,26,16,27,41,53,14,25,39,51,40,52,62,64)(4,9,19,30,20,31,36,49,18,29,43,55,46,57,61,59)(13,35,38,60,17,42,45,63,37,24,48,50,44,28,54,56);(1,3,5,15,6,16,21,41,7,14,22,39,23,40,47,62)(2,8,10,26,11,27,32,53,12,25,33,51,34,52,58,64)(4,17,43,54,20,37,61,38,18,44,19,45,46,13,36,48)(9,28,55,63,31,35,59,50,29,42,30,56,57,24,49,60);(1,4)(2,9)(3,13,14,37)(5,19)(6,20)(7,18)(8,24,25,35)(10,30)(11,31)(12,29)(15,38,39,48)(16,17,40,44)(21,36)(22,43)(23,46)(26,50,51,60)(27,28,52,42)(32,49)(33,55)(34,57)(41,45,62,54)(47,61)(53,56,64,63)(58,59)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93713, #93714, #143170, … |
| maximal normal subgroup of | #93713, #93714, #143170, … |
| maximal quotient of | #140560, #140564, #140582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6186);