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 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,6)(3,9)(4,13)(14,16)(18,22)(19,26)(24,28)(27,32)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,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×#55769, #8102975, #8102997, … |
| maximal normal subgroup of | C2×#55769, #8102975, #8102997, … |
| maximal quotient of | #586203, #587227, #587235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48757);