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 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,24,20,21)(22,30,31,25)(27,29,28,32);(1,2)(3,6)(4,5)(7,8)(10,12)(15,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(2,5)(6,8)(9,11,14,13)(10,15,12,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,13,16)(17,21,23,24)(18,20,26,19)(22,29,31,32)(25,28,30,27);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,15,12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=fg,e^2=f,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×#52616, #8092295, #8092472, … |
| maximal normal subgroup of | C2×#52616, #8092295, #8092472, … |
| maximal quotient of | #582313, #582520, #582565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45604);