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 | #3487 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,24)(13,18)(14,19)(15,28)(16,23)(17,25)(21,36)(22,32)(26,33)(27,31)(29,34)(30,37)(35,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,27)(14,23)(15,22)(17,30)(18,32)(19,20)(24,40)(25,39)(26,38)(28,31)(29,37)(33,36)(34,35);(9,12,14,26)(10,17,19,34)(11,21,23,38)(13,25,15,29)(16,30,20,35)(18,33,28,24)(22,37,27,39)(31,40,32,36);(3,8)(5,7)(9,13,14,15)(10,18,19,28)(11,22,23,27)(12,25,26,29)(16,31,20,32)(17,33,34,24)(21,37,38,39)(30,40,35,36);(10,19)(11,23)(17,34)(18,28)(21,38)(22,27)(24,33)(37,39)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=h,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 | #7054912, #7054922, #7054928, … |
| maximal normal subgroup of | #7054912, #7054922, #7054928, … |
| maximal quotient of | #6379083, #6379315, #6379718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54178);