Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1025 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,30)(23,32)(25,31)(27,34)(29,33)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11,14,25,15,23,13,24)(10,16,18,32,20,31,19,21)(12,22,28,38,29,36,27,37)(17,30,26,39,34,40,33,35);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,29,14,28)(10,17,19,33,20,34,18,26)(11,22,24,37,23,36,25,38)(16,30,21,35,31,40,32,39);(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38);(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,27,29,28)(16,32,31,21)(17,26,34,33)(22,37,36,38)(30,39,40,35)Polycyclic presentation:
2^8|b^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129970, #262986, #262988, … |
| maximal normal subgroup of | #129970, #262986, #262988, … |
| maximal quotient of | #6371043, #6371684, #6371688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53600);