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 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,12,14)(11,15,16,13)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,10)(6,9)(11,16)(13,15)(18,23)(20,27)(25,29)(28,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3,5,10)(2,6,8,9)(4,11,12,16)(7,13,14,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fh,e^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×#57757, #8101914, #8101963, … |
| maximal normal subgroup of | C2×#57757, #8101914, #8101963, … |
| maximal quotient of | #586926, #587076, #587132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50745);