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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(11,12)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,2)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(18,22)(19,25)(20,29)(30,32);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,16,13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,16,12,15)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|b^2=g,c^2=fgh,d^2=g,e^2=f,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×#56479, #8099274, #8099308, … |
| maximal normal subgroup of | C2×#56479, #8099274, #8099308, … |
| maximal quotient of | #586237, #587144, #587287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49467);