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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,23,26)(19,25,20,24)(21,31,32,22)(27,30,29,28);(1,2)(3,7)(4,5)(6,8)(17,19,23,20)(18,24,26,25)(21,27,32,29)(22,30,31,28);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,25,26,24)(21,27,32,29)(22,30,31,28);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,30)(25,28)(26,31);(1,3)(2,6)(4,8)(5,7)(17,22)(18,21)(19,28)(20,30)(23,31)(24,29)(25,27)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fg,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×#40815, #7639048, #7639579, … |
| maximal normal subgroup of | C2×#40815, #7639048, #7639579, … |
| maximal quotient of | #544616, #545148, #545613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33803);