Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | |
783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(10,11)(14,15)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31);(9,10,13,11)(12,14,16,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,27,24)(21,28,29,25)(26,30,32,31);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129843, #129844, #163737, … |
| maximal normal subgroup of | #129843, #129844, #163737, … |
| maximal quotient of | #516530, #516697, #516701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28312);