Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(1,2,4,6)(3,5,7,8)(10,12)(15,17)(16,20)(22,23)(26,31)(27,34)(29,38)(36,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98463, #251442, #251444, … |
| maximal normal subgroup of | #98463, #251442, #251444, … |
| maximal quotient of | #557045, #557052, #557389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39131);