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 | #3518 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
877 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)(6,8)(9,10,15,19)(11,16,23,32)(12,25,27,34)(13,20,14,18)(17,28,33,26)(21,35,37,40)(22,31,24,30)(29,38,39,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,29)(18,30)(19,32)(20,31)(25,35)(26,36)(27,37)(28,38)(33,39)(34,40);(9,12,15,27)(10,17,19,33)(11,21,23,37)(13,26,14,28)(16,29,32,39)(18,25,20,34)(22,36,24,38)(30,35,31,40);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,22,23,24)(12,26,27,28)(16,30,32,31)(17,25,33,34)(21,36,37,38)(29,35,39,40);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,33)(18,20)(21,37)(22,24)(25,34)(36,38)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,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 | #6358996, #6359000, #6359028, … |
| maximal normal subgroup of | #6358996, #6359000, #6359028, … |
| maximal quotient of | #6379601, #6379605, #6380403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54532);