Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,14)(12,16)(17,18,22,25)(19,23,27,21)(20,28,30,24)(26,29,32,31);(2,5)(6,8)(10,14)(11,15)(17,19,22,27)(18,21,25,23)(20,26,30,32)(24,29,28,31);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,29,23,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133476, #254528, #254530, … |
| maximal normal subgroup of | #133476, #254528, #254530, … |
| maximal quotient of | #553844, #554079, #554213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29807);