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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
744 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(20,28)(23,25)(26,32);(3,8)(4,6)(17,18,21,22)(19,23,27,25)(20,29,28,24)(26,31,32,30);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,13,14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,3)(2,4)(5,8)(6,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4)(2,3)(5,6)(7,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fg,d^2=f,e^2=f,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×#48862, #7704928, #7704956, … |
| maximal normal subgroup of | C2×#48862, #7704928, #7704956, … |
| maximal quotient of | #560576, #560590, #561677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41850);