Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
738 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)(13,16)(17,18,22,23)(19,27,28,24)(20,25,26,21)(29,32,31,30);(1,2)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,28)(21,25)(29,31);(2,3)(6,7)(9,11,12,15)(10,13,14,16)(17,19,22,28)(18,24,23,27)(20,31,26,29)(21,30,25,32);(1,3,5,2)(4,7,8,6)(17,20,22,26)(18,25,23,21)(19,29,28,31)(24,30,27,32);(1,4)(2,6)(3,7)(5,8)(17,21,22,25)(18,26,23,20)(19,30,28,32)(24,29,27,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132329, #253517, #253574, … |
| maximal normal subgroup of | #132329, #253517, #253574, … |
| maximal quotient of | #560389, #560525, #561586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43704);