Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(12,20)(15,22)(18,24)(21,23)(26,30)(28,36)(31,33)(34,40);(3,8)(5,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7206512, #7206516, #7206520, … |
| maximal normal subgroup of | #7206512, #7206516, #7206520, … |
| maximal quotient of | #582157, #582375, #582376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46060);