Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,5,8)(9,10,13,14)(11,15,16,12)(18,22)(20,28)(23,25)(26,32);(2,7)(3,5)(9,11,13,16)(10,12,14,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3,6,5)(2,8,7,4)(9,12)(10,11)(13,15)(14,16);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5,6,3)(2,4,7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=fh,e^2=g,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 | #97179, #130097, #133469, … |
| maximal normal subgroup of | #97179, #130097, #133469, … |
| maximal quotient of | #557222, #557538, #557682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39193);