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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
631 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)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(4,12)(8,13)(11,15)(14,16)(19,24)(21,30)(26,27)(28,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fgh,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×#56756, #8100074, #8100301, … |
| maximal normal subgroup of | C2×#56756, #8100074, #8100301, … |
| maximal quotient of | #586215, #586408, #587119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49744);