Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2,5,6)(3,4,7,8)(10,11)(14,15)(18,23)(20,27)(25,29)(28,32);(2,6)(3,7)(9,10,13,11)(12,14,16,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4)(2,3)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#55082, #8100705, #8101193, … |
| maximal normal subgroup of | C2×#55082, #8100705, #8101193, … |
| maximal quotient of | #585684, #586233, #586920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48070);