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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28);(2,5)(3,7)(17,19,22,24)(18,23,26,20)(21,30,31,27)(25,28,29,32);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,32)(28,29)(30,31);(1,2)(3,6)(4,5)(7,8)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=f,e^2=fg,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×#49928, #7705619, #7705676, … |
| maximal normal subgroup of | C2×#49928, #7705619, #7705676, … |
| maximal quotient of | #560666, #560950, #561750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42916);