Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #418 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1804 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 189 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)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,56)(17,32,48,40)(19,51)(21,54)(24,38)(27,55)(29,37)(30,43,39,53)(31,49,36,58)(33,59)(34,52)(35,46)(44,62,50,64)(47,61,57,63);(1,3,6,17,7,16,25,48)(2,8,28,40,10,13,12,32)(4,14,29,50,19,42,21,44)(5,23,26,58,27,15,46,49)(9,30,33,61,34,39,38,63)(11,36,55,56,35,31,22,45)(18,41,37,64,51,60,54,62)(20,53,59,47,52,43,24,57);(1,4)(2,9)(3,14)(5,24)(6,29)(7,19)(8,30)(10,34)(11,37)(12,38)(13,39)(15,47)(16,42)(17,50)(18,22)(20,26)(21,25)(23,57)(27,59)(28,33)(31,62)(32,63)(35,54)(36,64)(40,61)(41,45)(43,49)(44,48)(46,52)(51,55)(53,58)(56,60)Polycyclic presentation:
2^8|a^2=f,b^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12829, #493798, #494050, … |
| maximal normal subgroup of | C2×#12829, #493798, #494050, … |
| maximal quotient of | #134392, #134417, #134477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5817);