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 | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,23)(13,18)(14,27)(15,19)(17,24)(20,35)(21,30)(22,32)(25,33)(26,31)(28,34)(29,37)(36,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,22)(10,16,19,32)(12,20,25,38)(13,26,14,21)(17,29,34,39)(18,31,27,30)(23,35,33,40)(24,37,28,36);(1,3,5,4)(2,7,6,8)(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)(30,35)(31,40)(32,39);(1,4,5,3)(2,7,6,8);(9,13,15,14)(10,18,19,27)(11,21,22,26)(12,24,25,28)(16,30,32,31)(17,33,34,23)(20,36,38,37)(29,35,39,40)Polycyclic presentation:
2^8|b^2=fh,c^2=f,d^2=f,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 | #131080, #131083, #131888, … |
| maximal normal subgroup of | #131080, #131083, #131888, … |
| maximal quotient of | #6379463, #6380289, #6380654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54399);