Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
678 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,3,5,4)(2,6,7,8)(10,14)(15,20)(16,21)(22,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(2,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,4,5,3)(2,8,7,6)(11,17)(15,20)(18,23)(22,24)(26,32)(28,36)(30,38)(33,35);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133247, #251821, #251822, … |
| maximal normal subgroup of | #133247, #251821, #251822, … |
| maximal quotient of | #544276, #544280, #544289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29688);