Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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)(3,5)(4,8)(6,7)(9,10,13,12)(11,15,16,14)(18,19)(21,29)(23,25)(27,32);(1,3)(2,5)(4,6)(7,8)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,7)(3,4)(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,4,6,3)(2,8,7,5)(9,10,13,12)(11,14,16,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,5,6,8)(2,3,7,4)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50597, #7706031, #7706233, … |
| maximal normal subgroup of | C2×#50597, #7706031, #7706233, … |
| maximal quotient of | #560829, #560977, #561938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43585);