Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(11,15)(13,16)(17,18)(19,21)(20,28)(22,23)(24,27)(25,30)(26,31)(29,32);(9,10,12,14)(11,16,15,13)(18,23)(21,24)(25,28)(29,31);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,29)(30,32);(17,20,22,30)(18,25,23,28)(19,26,27,32)(21,29,24,31);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,24)(18,19,23,27)(20,29,30,31)(25,26,28,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=h,c^2=h,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6555882, #6555884, #6555885, … |
| maximal normal subgroup of | #6555882, #6555884, #6555885, … |
| maximal quotient of | #521487, #522139, #522651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32797);