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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,14,12)(11,15,13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29);(1,2,5,7)(3,4,8,6)(10,12)(11,13)(19,26)(22,25)(27,32)(29,31);(1,3,5,8)(2,6,7,4)(9,11,14,13)(10,15,12,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,4)(2,3)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=gh,d^2=f,e^2=fg,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×#57464, #8100705, #8101197, … |
| maximal normal subgroup of | C2×#57464, #8100705, #8101197, … |
| maximal quotient of | #585628, #585782, #587176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50452);