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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(4,12)(7,14)(10,16)(13,15)(18,23)(20,27)(25,29)(28,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=fgh,d^2=h,e^2=fg,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×#58775, #8104959, #8104978, … |
| maximal normal subgroup of | C2×#58775, #8104959, #8104978, … |
| maximal quotient of | #586144, #586357, #586461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51763);