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:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,34,39,37)(35,42,36,40)(38,45,46,41)(43,47,44,48);(2,6)(3,10)(8,12)(11,16)(18,22)(19,25)(20,29)(30,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=fg,d^2=fgh,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×#58567, #8104319, #8104427, … |
| maximal normal subgroup of | C2×#58567, #8104319, #8104427, … |
| maximal quotient of | #586225, #586622, #586956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51555);