Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
765 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)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,24,27,31)(20,28,30,21)(25,26,29,32);(1,2,5,4)(3,6,7,8)(10,13)(14,15)(18,23)(21,28)(24,31)(26,32);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,26)(22,27)(23,31)(25,30)(28,32);(9,11)(10,14)(12,16)(13,15)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29);(1,4,5,2)(3,8,7,6)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41732, #7639647, #7640241, … |
| maximal normal subgroup of | C2×#41732, #7639647, #7640241, … |
| maximal quotient of | #544799, #545250, #545759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34720);