Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,25)(19,23,20,27)(21,29,30,24)(26,32,28,31);(1,2)(3,5)(4,6)(7,8)(10,14)(12,15)(17,19,22,20)(18,23,25,27)(21,26,30,28)(24,31,29,32);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,12,13,15)(10,11,14,16)(17,20,22,19)(18,23,25,27)(21,28,30,26)(24,31,29,32);(1,3,4,7)(2,5,6,8)(17,21,22,30)(18,24,25,29)(19,26,20,28)(23,31,27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=h,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7227787, #7227791, #7227804, … |
| maximal normal subgroup of | #7227787, #7227791, #7227804, … |
| maximal quotient of | #582236, #582454, #582455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46121);