Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29);(2,5)(3,8)(9,10)(11,12)(13,14)(15,16)(17,19,23,20)(18,24,26,21)(22,28,30,27)(25,29,32,31);(2,5)(6,7)(10,14)(11,15)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,14)(13,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=fg,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58059, #8104596, #8104687, … |
| maximal normal subgroup of | C2×#58059, #8104596, #8104687, … |
| maximal quotient of | #585275, #585739, #586707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51047);