Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13717 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #2233 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2233 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12)(3,13,14,42)(4,18,19,52)(5,24,27,60)(6,28,26,11)(8,30,31,15)(9,34,35,22)(10,38,40,21)(16,50,46,64)(17,51,29,45)(20,56,48,39)(23,43,55,54)(25,44,61,36)(32,57,63,59)(33,49,41,62)(37,58,53,47);(4,19)(9,35)(13,42)(15,30)(18,52)(20,48)(22,34)(23,55)(36,44)(37,53)(43,54)(45,51)(47,58)(49,62)(50,64)(57,59);(1,3,6,29,7,14,26,17)(2,8,11,41,12,31,28,33)(4,20,23,59,19,48,55,57)(5,25,21,16,27,61,38,46)(9,36,37,50,35,44,53,64)(10,39,24,32,40,56,60,63)(13,43,51,18,42,54,45,52)(15,47,62,22,30,58,49,34)Polycyclic presentation:
2^8|a^2=h,c^2=dg,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^b=eh,d^c=dh,e^c=efh,f^c=fg,g^c=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151491, #151492, #151494, … |
| maximal normal subgroup of | #151491, #151492, #151494, … |
| maximal quotient of | #152373, #152429, #152518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6706);