Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,49)(8,22,27,40)(9,28,32,47)(10,33,30,15)(12,35,38,19)(13,24,42,25)(14,43,45,31)(18,50,53,66)(20,54,34,44)(23,48,51,39)(26,62,59,56)(29,46,64,37)(36,57,65,52)(41,61,55,60)(58,67,63,68);(5,7,11,21)(6,12,16,38)(8,23,27,51)(9,29,32,64)(10,34,30,20)(13,39,42,48)(14,44,45,54)(15,46,33,37)(17,47,49,28)(18,24,53,25)(19,43,35,31)(22,36,40,65)(26,63,59,58)(41,68,55,67)(50,56,66,62)(52,61,57,60);(2,4)(5,8,9,24,30,59,43,61,11,27,32,25,10,26,31,60)(6,13,14,40,33,55,47,56,16,42,45,22,15,41,28,62)(7,18,19,51,34,57,64,58,21,53,35,23,20,52,29,63)(12,36,17,48,46,66,54,67,38,65,49,39,37,50,44,68)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93815, #93817, #151291, … |
| maximal normal subgroup of | #93815, #93817, #151291, … |
| maximal quotient of | #145362, #145394, #145449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6429);