Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/32) (order 4096) · Inn(G) = #2215 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(2,9)(3,16)(4,10)(5,7)(6,20)(11,37)(12,31)(13,43)(14,29)(15,32)(17,47)(18,49)(19,24)(21,35)(22,34)(23,36)(25,26)(27,38)(28,30)(33,45)(39,62)(40,55)(41,54)(42,57)(44,56)(48,59)(50,58)(52,61)(53,60)(63,64);(1,2,10,30,6,13,21,29,5,12,34,56,25,42,23,32,8,15,36,57,26,44,22,31,7,14,35,43,20,28,4,9)(3,11,33,53,18,39,50,55,24,41,59,64,51,63,48,54,19,40,58,62,49,60,45,37,16,38,47,61,46,52,17,27);(1,3)(2,11)(4,17)(5,24)(6,18)(7,16)(8,19)(9,27)(10,33)(12,41)(13,39)(14,38)(15,40)(20,46)(21,50)(22,45)(23,48)(25,51)(26,49)(28,52)(29,55)(30,53)(31,37)(32,54)(34,59)(35,47)(36,58)(42,63)(43,61)(44,60)(56,64)(57,62)Polycyclic presentation:
2^8|b^2=df,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #153609, #153610, #153664, … |
| maximal normal subgroup of | #153609, #153610, #153664, … |
| maximal quotient of | #151637, #151640, #151646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6688);