Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
760 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)(11,15)(14,16)(17,18,22,23)(19,26,29,24)(20,25,30,21)(27,28,32,31);(9,10)(11,14)(12,13)(15,16)(18,23)(20,30)(24,26)(27,32);(2,5)(6,7)(10,13)(14,16)(17,19,22,29)(18,24,23,26)(20,27,30,32)(21,28,25,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,31)(26,28)(29,32);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,15)(13,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52635, #8092148, #8092479, … |
| maximal normal subgroup of | C2×#52635, #8092148, #8092479, … |
| maximal quotient of | #582312, #582518, #582563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45623);