Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #6344638 (order 512) · Inn(G) = C2×D4 · Out(G) = #131 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | D4×C16 |
| prime radical | #135 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,7,14,31,46,8,12,32,44,34,45,37,62,9,15,33,47,35,17,49,51,5,27,29,58,30,50,55,61)(3,16,19,48,20,25,53,57,18,10,36,38,52,26,11,41,21,39,43,63,54,23,42,56,28,40,59,64,60,4,22,24);(1,3)(2,10)(4,17)(5,28)(6,19)(7,20)(8,18)(9,21)(11,37)(12,39)(13,38)(14,26)(15,40)(16,27)(22,55)(23,45)(24,51)(25,50)(29,59)(30,60)(31,53)(32,36)(33,43)(34,52)(35,54)(41,46)(42,49)(44,63)(47,64)(48,58)(56,62)(57,61);(1,4,6,24,7,16,31,48,8,25,32,57,34,10,37,38,9,26,33,41,35,39,49,63,5,23,29,56,30,40,55,64)(2,11,13,21,14,43,46,54,12,42,44,28,45,59,62,60,15,22,47,3,17,19,51,20,27,53,58,18,50,36,61,52)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch,d^a=dh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #151103, #151682, #151683, … |
| maximal normal subgroup of | #151103, #151682, #151683, … |
| maximal quotient of | #151074, #151088, #151089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6636);