Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| 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 |
| prime radical | #3013 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
998 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)(9,10,14,17)(11,15,20,29)(12,22,24,32)(13,25,27,18)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,29)(18,30)(22,33)(23,35)(24,34)(25,37)(26,38)(27,36)(31,39)(32,40);(9,12,14,24)(10,16,17,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(18,32,25,22)(21,38,36,35)(30,40,37,33);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,31)(18,25)(22,32)(28,39)(30,37)(33,40);(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6439759, #6439761, #6439762, … |
| maximal normal subgroup of | #6439759, #6439761, #6439762, … |
| maximal quotient of | #6370989, #6371799, #6371800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53565);