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 | |
681 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)(9,10,14,15)(11,12,17,20)(13,21,22,16)(18,24,23,19)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,2,4,6)(3,5,7,8)(10,15)(11,17)(13,22)(19,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,19,20,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=h,b^2=g,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 | #98467, #251435, #251437, … |
| maximal normal subgroup of | #98467, #251435, #251437, … |
| maximal quotient of | #557045, #557052, #557388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39128);