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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(4,12)(8,13)(11,15)(14,16)(18,20)(19,25)(21,29)(30,32)(34,39)(36,40)(42,45)(44,47);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^8|a^2=h,c^2=fgh,d^2=fg,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×#56761, #8099538, #8100102, … |
| maximal normal subgroup of | C2×#56761, #8099538, #8100102, … |
| maximal quotient of | #586247, #586335, #586536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49749);