Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
939 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)(11,16)(12,24)(13,18)(14,28)(15,19)(17,26)(20,33)(21,30)(22,31)(23,37)(25,32)(27,38)(29,36)(34,40)(35,39);(2,5)(6,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,33)(25,35)(26,36)(27,34)(28,37)(32,39)(38,40);(9,12,14,26,15,27,13,25)(10,17,18,24,19,32,28,38)(11,20,23,36,21,34,22,35)(16,29,31,33,30,39,37,40);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,34)(22,23)(29,39)(31,37)(33,40)(35,36);(9,13,15,14)(10,18,19,28)(11,22,21,23)(12,25,27,26)(16,31,30,37)(17,24,32,38)(20,35,34,36)(29,33,39,40)Polycyclic presentation:
2^8|c^2=g,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130031, #262981, #262984, … |
| maximal normal subgroup of | #130031, #262981, #262984, … |
| maximal quotient of | #6371175, #6372101, #6372113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53670);