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)(10,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(2,6)(7,12)(8,13)(14,16)(18,19)(20,27)(24,26)(28,32)(33,35,38,36)(34,39,42,40)(37,44,46,43)(41,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=f,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×#56001, #8102735, #8103198, … |
| maximal normal subgroup of | C2×#56001, #8102735, #8103198, … |
| maximal quotient of | #585748, #587056, #587129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48989);