Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,24)(13,17)(14,19)(15,20)(16,30)(18,28)(22,36)(23,33)(25,38)(26,34)(27,39)(29,32)(31,40)(35,37);(2,5)(6,8)(9,11,15,13)(10,17,20,21)(12,22,27,25)(14,29,16,23)(18,31,34,35)(19,33,30,32)(24,38,39,36)(26,40,28,37);(9,12,16,28,15,27,14,26)(10,18,19,24,20,34,30,39)(11,22,23,37,13,25,29,40)(17,31,33,38,21,35,32,36);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,20,21)(12,25,27,22)(14,23,16,29)(18,31,34,35)(19,33,30,32)(24,38,39,36)(26,37,28,40);(9,14,15,16)(10,19,20,30)(11,23,13,29)(12,26,27,28)(17,32,21,33)(18,24,34,39)(22,37,25,40)(31,36,35,38)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130030, #130049, #482711, … |
| maximal normal subgroup of | #130030, #130049, #482711, … |
| maximal quotient of | #6371188, #6372131, #6372132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53705);