Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/32) (order 12288) · Inn(G) = C22×D16 · Out(G) = C2×D3×C8 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9)(4,8)(5,22)(6,25)(7,13)(10,18)(11,19)(12,23)(14,30)(15,26)(16,44)(17,45)(20,37)(21,29)(24,51)(27,47)(28,46)(31,41)(32,42)(33,40)(34,50)(35,48)(36,49)(38,52)(39,62)(43,58)(53,63)(54,61)(55,59)(56,60)(57,64);(2,8)(4,18)(5,19)(6,23)(9,26)(10,11)(12,37)(13,29)(15,41)(16,42)(17,40)(20,50)(21,48)(22,27)(24,49)(25,28)(30,52)(31,32)(33,58)(34,51)(35,36)(38,59)(39,60)(43,64)(44,53)(45,54)(46,47)(55,56)(57,62)(61,63);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,26)(10,31)(11,32)(12,33)(13,30)(18,41)(19,42)(20,43)(21,38)(22,44)(23,40)(24,39)(25,45)(27,53)(28,54)(29,52)(34,57)(35,55)(36,56)(37,58)(46,61)(47,63)(48,59)(49,60)(50,64)(51,62);(3,14)(9,30)(15,38)(16,39)(17,40)(26,52)(31,55)(32,56)(33,45)(41,59)(42,60)(43,61)(44,62)(53,57)(54,58)(63,64)Polycyclic presentation:
2^8|e^2=fg,f^2=gh,g^2=h,b^a=be,e^a=ef,f^a=fg,g^a=gh,e^b=ef,f^b=fg,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33982⋊C3 |
| maximal subgroup of | #153609, #153611, #513188, … |
| maximal normal subgroup of | #153609, #153611, #513188, … |
| maximal quotient of | #512691, #512719, #512764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26970);