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:
(2,5)(3,7)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(29,37)(31,38)(34,40)(36,39);(2,5)(6,8)(10,11)(12,19)(16,18)(20,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,37)(26,31,32,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^8|b^2=gh,c^2=h,d^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 | #7232340, #7232344, #7232356, … |
| maximal normal subgroup of | #7232340, #7232344, #7232356, … |
| maximal quotient of | #582229, #582439, #582440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46141);