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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(18,19)(21,29)(23,25)(27,32);(9,10)(11,14)(12,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,3,4,8)(2,6,5,7)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=h,d^2=g,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×#50256, #7705935, #7706219, … |
| maximal normal subgroup of | C2×#50256, #7705935, #7706219, … |
| maximal quotient of | #561208, #561308, #562403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43244);