Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12463 |
| automorphism | Aut(G) = #72309687 (order 2048) · Inn(G) = #1675 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1675 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,15,21,47)(4,22)(5,27,29,12)(6,11,16,42)(7,31,28,13)(9,32,36,18)(10,37)(17,50)(19,45,48,35)(20,34,55,46)(23,41)(24,58)(25,43)(26,38)(30,39)(33,52)(40,49)(44,59)(51,57)(53,61)(54,60)(56,63)(62,64);(1,3,16,48,7,20,5,18,8,21,6,19,28,55,29,32)(2,9,12,34,13,35,11,15,14,36,27,46,31,45,42,47)(4,17,49,64,24,52,23,51,26,54,30,56,25,53,37,59)(10,33,43,62,39,50,38,44,41,61,58,63,40,60,22,57);(1,4)(2,10)(3,17)(5,23)(6,30)(7,24)(8,26)(9,33)(11,38)(12,43)(13,39)(14,41)(15,44)(16,49)(18,51)(19,56)(20,52)(21,54)(22,42)(25,28)(27,58)(29,37)(31,40)(32,59)(34,62)(35,50)(36,61)(45,60)(46,63)(47,57)(48,64)(53,55)Polycyclic presentation:
2^8|a^2=fg,b^2=dfh,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^c=egh,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151578, #151579, #151580, … |
| maximal normal subgroup of | #151578, #151579, #151580, … |
| maximal quotient of | #134973, #138571, #140261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5452);