Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,20)(12,25)(13,18)(14,19)(15,28)(16,23)(17,26)(21,36)(22,35)(24,31)(27,32)(29,33)(30,39)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,23)(10,16,19,20)(12,21,27,38)(13,22,15,24)(17,30,33,34)(18,31,28,35)(25,40,32,36)(26,37,29,39);(9,12)(10,17)(11,21)(13,26)(14,27)(15,29)(16,30)(18,32)(19,33)(20,34)(22,37)(23,38)(24,39)(25,28)(31,36)(35,40);(2,5)(6,8)(9,13,14,15)(10,18,19,28)(11,22,23,24)(12,26,27,29)(16,31,20,35)(17,32,33,25)(21,37,38,39)(30,36,34,40);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,33)(18,28)(21,38)(22,24)(25,32)(37,39)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6364525, #6364529, #6364621, … |
| maximal normal subgroup of | #6364525, #6364529, #6364621, … |
| maximal quotient of | #6386459, #6388579, #6388586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55057);