Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6)(7,17)(8,12)(9,25)(10,30)(11,32)(13,19)(14,28)(15,27)(16,24)(18,43)(20,50)(21,52)(22,44)(23,55)(26,41)(29,62)(31,39)(33,45)(34,46)(35,48)(36,51)(37,60)(38,59)(40,67)(42,49)(47,68)(53,66)(54,58)(56,65)(57,61)(63,64);(1,2,3,4)(6,12)(13,34)(14,35)(15,36)(16,32)(17,43)(25,44)(30,55)(37,64)(38,42)(39,65)(40,57)(41,66)(45,52)(50,68)(58,62);(5,7,8,18)(6,13,12,34)(9,26,22,53)(10,31,23,56)(11,33,24,21)(14,39,35,65)(15,41,36,66)(16,42,32,38)(17,44,43,25)(19,48,46,28)(20,51,47,27)(29,63,54,60)(30,37,55,64)(40,68,57,50)(45,58,52,62)(49,61,59,67)Polycyclic presentation:
2^8|b^2=dh,c^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13690, #512913, #512915, … |
| maximal normal subgroup of | C2×#13690, #512913, #512915, … |
| maximal quotient of | #151621, #151625, #151633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6678);