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 | |
629 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)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(4,12)(8,13)(11,15)(14,16)(18,21)(19,25)(20,28)(31,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,25)(18,23,21,27)(20,29,28,26)(24,32,30,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,28)(18,24,21,30)(19,26,25,29)(23,31,27,32)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,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×#55855, #8101701, #8101959, … |
| maximal normal subgroup of | C2×#55855, #8101701, #8101959, … |
| maximal quotient of | #586041, #586632, #587138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48843);