Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | Aut(G) = S4×#12067 (order 6144) · Inn(G) = C22×D16 · Out(G) = 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 | |
231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,9,14,30)(4,8,21,29)(5,22,24,51)(6,25,23,12)(10,34,36,18)(11,37,38,19)(15,26,40,52)(16,46,41,62)(17,47,42,33)(20,39,48,28)(27,50,35,49)(31,57,55,43)(32,59,56,44)(45,60,61,54)(53,64,58,63);(2,8)(4,18)(5,19)(6,23)(9,26)(10,11)(12,39)(13,29)(15,43)(16,44)(17,42)(20,50)(21,34)(22,27)(24,37)(25,28)(30,52)(31,32)(33,60)(35,51)(36,38)(40,57)(41,59)(45,64)(46,53)(47,54)(48,49)(55,56)(58,62)(61,63);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,26)(10,31)(11,32)(12,33)(13,30)(18,43)(19,44)(20,45)(21,40)(22,46)(23,42)(24,41)(25,47)(27,53)(28,54)(29,52)(34,57)(35,58)(36,55)(37,59)(38,56)(39,60)(48,61)(49,63)(50,64)(51,62);(3,14)(9,30)(15,40)(16,41)(17,42)(26,52)(31,55)(32,56)(33,47)(43,57)(44,59)(45,61)(46,62)(53,58)(54,60)(63,64)Polycyclic presentation:
2^8|a^2=h,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 | #33983⋊C3 |
| maximal subgroup of | #513190, #513191, #513193, … |
| maximal normal subgroup of | #513190, #513191, #513193, … |
| maximal quotient of | #512691, #512692, #512720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26971);