Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,7,13,8,14,6,12)(3,15,20,45,21,46,19,44)(4,22,25,10,26,38,24,39)(5,27,16,42,30,11,31,33)(9,32,35,28,36,54,41,18)(17,48,51,43,52,63,50,60)(23,57,47,37,29,56,58,40)(34,53,61,64,62,49,55,59);(1,3,16,32)(2,9,33,45)(4,17,47,59)(5,28,7,19)(6,20,30,18)(8,21,31,54)(10,34,56,48)(11,15,13,35)(12,41,27,46)(14,36,42,44)(22,55,40,43)(23,53,25,50)(24,51,29,49)(26,52,58,64)(37,60,38,61)(39,62,57,63);(1,4,8,26)(2,10,14,39)(3,17,21,52)(5,29,30,23)(6,25,7,24)(9,34,36,62)(11,40,27,37)(12,38,13,22)(15,43,46,60)(16,47,31,58)(18,53,28,49)(19,51,20,50)(32,59,54,64)(33,56,42,57)(35,55,41,61)(44,63,45,48)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13101, #494757, #494759, … |
| maximal normal subgroup of | C2×#13101, #494757, #494759, … |
| maximal quotient of | #137685, #137691, #137733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6089);