Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1964 |
| commutator | C22×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,13,18,36)(4,19)(5,24,14,10)(6,27)(7,12)(8,16)(9,22)(11,20)(15,39,43,62)(17,45,40,35)(21,50,37,47)(23,48)(25,30)(26,33,38,54)(28,57)(29,55)(31,52,53,61)(32,49)(34,59,60,42)(41,46)(44,56)(51,63)(58,64);(1,3,14,16,7,18,5,25)(2,8,24,36,12,30,10,13)(4,15,37,41,23,43,21,51)(6,17,38,44,20,40,26,55)(9,28,53,60,32,58,31,34)(11,29,54,45,27,56,33,35)(19,46,50,62,48,63,47,39)(22,42,61,57,49,59,52,64);(1,4,20,49,7,23,6,22)(2,9,27,48,12,32,11,19)(3,15,40,59,18,43,17,42)(5,21,38,61,14,37,26,52)(8,28,56,63,30,58,29,46)(10,31,54,50,24,53,33,47)(13,34,45,62,36,60,35,39)(16,41,55,64,25,51,44,57)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #138942, #138943, #138944, … |
| maximal normal subgroup of | #138942, #138943, #138944, … |
| maximal quotient of | #140616, #140635, #140645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6218);