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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
767 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)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,5)(6,8)(9,10,13,12)(11,14,15,16)(18,19)(23,25)(24,26)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,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 | #132018, #253369, #253371, … |
| maximal normal subgroup of | #132018, #253369, #253371, … |
| maximal quotient of | #553840, #554081, #554222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29825);