Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = #353 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1352 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,11,23,38)(3,13,15,39,16,40,45,62)(4,9,19,32,20,34,49,36)(6,12,24,37,25,31,53,59)(8,26,27,18,29,46,55,47)(14,28,42,35,43,56,44,61)(17,30,22,51,41,57,52,64)(21,33,48,60,50,58,54,63);(1,3)(2,8)(4,18,20,47)(5,22,23,52)(6,17)(7,16)(9,30,34,57)(10,35,38,61)(11,29)(12,28)(13,33,40,58)(14,21,43,50)(15,24,45,53)(19,42)(25,41)(26,54)(27,37,55,59)(31,56)(32,39)(36,62)(44,49)(46,48)(51,60)(63,64)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=fg,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,d^c=dg,e^c=eh,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7439, #150297, #150344, … |
| maximal normal subgroup of | C2×#7439, #150297, #150344, … |
| maximal quotient of | #93243, #93244, #93245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 427);