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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(14,16)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28);(2,5)(3,7)(9,10,13,12)(11,14,15,16)(18,23)(21,30)(24,27)(28,32);(1,2,4,5)(3,6,7,8)(9,11,13,15)(10,14,12,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32);(17,20,22,29)(18,24,23,27)(19,26,25,31)(21,28,30,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=f,e^2=g,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 | #6534828, #6534833, #6534837, … |
| maximal normal subgroup of | #6534828, #6534833, #6534837, … |
| maximal quotient of | #521682, #522306, #522785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31265);