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 | #2703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(2,6)(3,7)(9,10)(11,15)(12,14)(13,16)(18,22)(19,26)(24,28)(27,32);(9,10,12,14)(11,13,16,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,c^2=gh,d^2=f,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×#56554, #8099012, #8099238, … |
| maximal normal subgroup of | C2×#56554, #8099012, #8099238, … |
| maximal quotient of | #585872, #586583, #587207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49542);