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:
(1,2,5,8)(3,9,10,6)(4,7,12,14)(11,15,16,13)(17,18,22,23)(19,20,27,24)(21,29,30,25)(26,31,32,28)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(3,10)(6,9)(11,16)(13,15)(18,23)(20,24)(21,30)(26,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,9)(12,16)(14,15)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,46)(42,48)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=f,d^2=gh,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×#58536, #8104369, #8104430, … |
| maximal normal subgroup of | C2×#58536, #8104369, #8104430, … |
| maximal quotient of | #585906, #586693, #587292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51524);