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 | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,12)(11,15,16,14)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,5)(3,7)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32);(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=fg,b^2=f,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 | #6543667, #6543679, #6543685, … |
| maximal normal subgroup of | #6543667, #6543679, #6543685, … |
| maximal quotient of | #521676, #522306, #522795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31255);