Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,13,17)(11,19,21,15)(12,23,24,32)(14,27,26,18)(16,28,31,25)(20,35,36,40)(22,30,38,34)(29,37,33,39);(1,3,5,8)(2,6,4,7)(9,11,13,21)(10,15,17,19)(12,20,24,36)(14,22,26,38)(16,29,31,33)(18,30,27,34)(23,40,32,35)(25,37,28,39);(9,12,13,24)(10,16,17,31)(11,20,21,36)(14,28,26,25)(15,29,19,33)(18,32,27,23)(22,39,38,37)(30,35,34,40);(1,4,5,2)(3,6,8,7)(11,21)(15,19)(20,36)(22,38)(29,33)(30,34)(35,40)(37,39);(10,17)(11,21)(16,31)(18,27)(20,36)(22,38)(23,32)(37,39)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #7182409, #7182412, #7182421, … |
| maximal normal subgroup of | #7182409, #7182412, #7182421, … |
| maximal quotient of | #6379401, #6379404, #6380142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54374);