Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
1027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,27)(13,19)(14,20)(15,31)(16,25)(17,28)(18,29)(22,37)(23,33)(24,38)(26,34)(30,36)(32,40)(35,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,35)(18,33)(19,34)(20,21)(22,29)(27,37)(28,40)(30,39)(31,38)(32,36);(3,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,32)(18,19)(20,36)(21,35)(23,25)(24,39)(26,40)(27,31)(33,38)(34,37);(2,6)(5,7)(9,12)(10,18)(11,23)(13,30)(14,29)(15,28)(16,33)(17,19)(20,27)(21,37)(22,25)(24,39)(26,40)(31,36)(32,38)(34,35);(9,13,14,15)(10,19,20,31)(11,24,25,26)(12,28,29,30)(16,34,21,38)(17,18,36,27)(22,39,23,40)(32,37,35,33)Polycyclic presentation:
2^8|b^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152115, #512655, #512668, … |
| maximal normal subgroup of | #152115, #512655, #512668, … |
| maximal quotient of | #6374313, #6374466, #6374976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53811);