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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31);(2,3)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(10,14)(11,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,14)(13,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=fgh,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×#58201, #8104419, #8104428, … |
| maximal normal subgroup of | C2×#58201, #8104419, #8104428, … |
| maximal quotient of | #586068, #587186, #587329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51189);