Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | #525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,23)(13,19)(14,27)(15,18)(17,24)(20,35)(21,31)(22,30)(25,34)(26,32)(28,33)(29,37)(36,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,22)(10,16,18,30)(12,20,25,38)(13,26,14,21)(17,29,33,39)(19,32,27,31)(23,35,34,40)(24,37,28,36);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,20)(13,24)(14,28)(15,25)(16,29)(18,33)(19,34)(21,36)(22,38)(23,27)(26,37)(30,39)(31,35)(32,40);(3,8)(6,7)(10,18)(16,30)(17,33)(19,27)(23,34)(29,39)(31,32)(35,40);(9,13,15,14)(10,19,18,27)(11,21,22,26)(12,24,25,28)(16,31,30,32)(17,34,33,23)(20,36,38,37)(29,35,39,40)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,d^b=df,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130849, #500008, #500356, … |
| maximal normal subgroup of | #130849, #500008, #500356, … |
| maximal quotient of | #6379308, #6379856, #6379863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54257);