Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12463 |
| automorphism | Aut(G) = #72309663 (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)(3,15,20,47)(4,21)(5,26,28,12)(6,11,30,42)(7,31,27,13)(9,32,36,17)(10,37)(16,48)(18,45,55,35)(19,34,53,46)(22,41)(23,57)(24,43)(25,38)(29,39)(33,50)(40,58)(44,59)(49,56)(51,61)(52,60)(54,63)(62,64);(1,3,6,18,7,19,28,32,8,20,30,55,27,53,5,17)(2,9,26,46,13,35,42,47,14,36,12,34,31,45,11,15)(4,16,29,54,23,50,37,59,25,52,58,64,24,51,22,49)(10,33,57,63,39,48,21,56,41,61,43,62,40,60,38,44);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,25)(9,33)(11,38)(12,43)(13,39)(14,41)(15,44)(17,49)(18,54)(19,50)(20,52)(21,42)(24,27)(26,57)(28,37)(30,58)(31,40)(32,59)(34,62)(35,48)(36,61)(45,60)(46,63)(47,56)(51,53)(55,64)Polycyclic presentation:
2^8|a^2=fg,b^2=df,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 | #151666, #151667, #151682, … |
| maximal normal subgroup of | #151666, #151667, #151682, … |
| maximal quotient of | #134972, #138571, #140261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5451);