Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
779 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,30)(26,32)(29,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,22,23,30)(20,26,28,31)(24,29,27,32);(10,14)(13,16)(18,23)(19,25)(24,27)(26,31);(17,20,21,28)(18,24,23,27)(19,26,25,31)(22,29,30,32);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #207462, #207478, #207492, … |
| maximal normal subgroup of | #207462, #207478, #207492, … |
| maximal quotient of | #515049, #515066, #515077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27617);