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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1786 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 85 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)(3,14)(4,19,22,37)(5,23)(6,12)(7,29)(8,13)(9,16)(10,21,36,28)(11,26)(15,43,46,63)(17,41)(18,42)(20,35,53,61)(24,32)(25,39)(27,55)(30,38)(31,58,59,51)(33,56)(34,48)(40,45,62,47)(44,49,64,60)(50,57,54,52);(1,3)(2,9)(4,15)(5,24)(6,17)(7,30)(8,18)(10,31)(11,38)(12,32)(13,34)(14,23)(16,26)(19,49)(20,51)(21,45)(22,46)(25,48)(27,56)(28,47)(29,33)(35,43)(36,59)(37,60)(39,41)(40,54)(42,55)(44,52)(50,62)(53,58)(57,64)(61,63);(1,4)(2,10)(3,15)(5,20)(6,28)(7,21)(8,22)(9,31)(11,35)(12,37)(13,36)(14,40)(16,44)(17,47)(18,46)(19,29)(23,54)(24,51)(25,57)(26,52)(27,53)(30,45)(32,60)(33,49)(34,59)(38,43)(39,61)(41,63)(42,62)(48,64)(50,55)(56,58)Polycyclic presentation:
2^8|a^2=f,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,g^b=gh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12747, #493547, #495719, … |
| maximal normal subgroup of | C2×#12747, #493547, #495719, … |
| maximal quotient of | #129520, #129521, #129563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5735);