Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| 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 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10,12,16)(11,21,23,44)(13,30,27,57)(14,33,28,58)(15,34,29,20)(17,38,35,24)(18,41,36,22)(19,43,37,31)(25,55,51,72)(26,56,52,47)(32,61,49,60)(39,67,63,71)(40,54,62,66)(42,50,59,68)(45,65,69,48)(46,64,70,53);(1,3,4,8)(2,5,6,7)(10,16)(17,35)(18,36)(19,37)(20,34)(21,44)(30,57)(33,58)(39,63)(40,62)(42,59)(45,69)(46,70)(47,56)(55,72)(60,61);(9,11,22,48,14,25,49,66,29,52,68,67,43,64,27,38,12,23,41,65,28,51,32,54,15,26,50,71,31,53,13,24)(10,17,30,46,19,39,59,56,34,62,61,72,58,69,36,44,16,35,57,70,37,63,42,47,20,40,60,55,33,45,18,21)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #94742, #94744, #151666, … |
| maximal normal subgroup of | #94742, #94744, #151666, … |
| maximal quotient of | #152403, #152851, #152852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6720);