Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | #3045 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,7)(9,10)(11,20)(12,24)(13,18)(14,19)(15,28)(16,22)(17,25)(21,36)(23,32)(26,33)(27,31)(29,34)(30,38)(35,39)(37,40);(1,2,4,5)(3,7,8,6)(9,11,14,22)(10,16,19,20)(12,21,26,37)(13,27,15,23)(17,30,34,35)(18,32,28,31)(24,40,33,36)(25,39,29,38);(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,28)(27,39)(31,40)(32,36);(1,3,4,8)(2,6,5,7)(11,22)(16,20)(21,37)(23,27)(30,35)(31,32)(36,40)(38,39);(9,13,14,15)(10,18,19,28)(11,23,22,27)(12,25,26,29)(16,31,20,32)(17,33,34,24)(21,38,37,39)(30,40,35,36)Polycyclic presentation:
2^8|b^2=f,d^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #220608, #220609, #220624, … |
| maximal normal subgroup of | #220608, #220609, #220624, … |
| maximal quotient of | #6379393, #6379464, #6380334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54319);