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 | #3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
873 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)(6,7)(9,10,14,19)(11,21,23,16)(12,25,27,34)(13,20,15,18)(17,29,32,26)(22,36,37,40)(24,31,28,33)(30,39,35,38);(1,2,4,5)(3,7,8,6)(9,11,14,23)(10,16,19,21)(12,22,27,37)(13,28,15,24)(17,30,32,35)(18,33,20,31)(25,40,34,36)(26,39,29,38);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,30)(18,25)(19,32)(20,34)(21,35)(23,37)(24,38)(28,39)(31,36)(33,40);(1,3,4,8)(2,6,5,7)(11,23)(16,21)(22,37)(24,28)(30,35)(31,33)(36,40)(38,39);(9,13,14,15)(10,18,19,20)(11,24,23,28)(12,26,27,29)(16,31,21,33)(17,25,32,34)(22,38,37,39)(30,36,35,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fh,e^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 | #7165840, #7165842, #7165851, … |
| maximal normal subgroup of | #7165840, #7165842, #7165851, … |
| maximal quotient of | #6379404, #6379472, #6380341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54338);