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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
656 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)(19,25,27,23)(20,28,29,24)(26,31,30,32)(34,38)(35,41)(36,44)(46,48);(2,6)(3,9)(4,13)(14,16)(18,22)(20,29)(23,25)(26,30)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=fg,d^2=g,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×#55887, #8102854, #8103186, … |
| maximal normal subgroup of | C2×#55887, #8102854, #8103186, … |
| maximal quotient of | #585901, #586419, #586645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48875);