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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,12)(7,13,14,8)(9,11,17,18)(15,19,20,16)(21,22)(23,30)(24,29)(25,28)(26,27)(31,36)(32,35)(33,34);(7,14)(8,13)(9,17)(11,18)(21,23,27,25)(22,28,26,30)(24,32,34,31)(29,36,33,35);(1,2)(3,4)(5,7,10,14)(6,8,12,13)(9,15,17,20)(11,16,18,19)(21,24,27,34)(22,29,26,33)(23,31,25,32)(28,35,30,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,19)(18,20)(21,25,27,23)(22,28,26,30)(24,31,34,32)(29,36,33,35);(5,9)(6,11)(7,15)(8,16)(10,17)(12,18)(13,19)(14,20)(21,26,27,22)(23,28,25,30)(24,33,34,29)(31,35,32,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,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×#50181, #7706124, #7706177, … |
| maximal normal subgroup of | C2×#50181, #7706124, #7706177, … |
| maximal quotient of | #560332, #560352, #561427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43169);