Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,3)(4,7,5,8)(10,13)(14,16)(17,18,22,23)(19,21,25,30)(20,27,29,24)(26,32,31,28);(1,3,6,2)(4,7,5,8)(18,23)(21,30)(24,27)(28,32);(1,4,6,5)(2,7,3,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,21,23,30)(20,26,29,31)(24,28,27,32);(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,27)(25,31)(30,32);(1,5,6,4)(2,8,3,7)(17,21)(18,25)(19,23)(20,28)(22,30)(24,31)(26,27)(29,32)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39775, #7637338, #7637366, … |
| maximal normal subgroup of | C2×#39775, #7637338, #7637366, … |
| maximal quotient of | #521508, #522159, #522663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32763);