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 | |
814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,14,16,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(18,23)(19,26)(25,28)(27,32);(2,5)(6,7)(10,11)(14,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=fg,e^2=f,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×#40804, #7639422, #7639582, … |
| maximal normal subgroup of | C2×#40804, #7639422, #7639582, … |
| maximal quotient of | #544722, #545210, #545725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33792);