Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(2,6)(3,8)(9,10,12,14)(11,13,16,15)(18,22)(23,25)(24,28)(30,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,e^2=fh,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 | #131588, #131589, #484775, … |
| maximal normal subgroup of | #131588, #131589, #484775, … |
| maximal quotient of | #553718, #554015, #554036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30369);