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 | |
737 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)(9,10,13,11)(12,15,16,14)(19,23)(20,27)(21,29)(24,31);(1,2)(3,5)(4,6)(7,8)(10,11)(12,16)(17,18,22,25)(19,20,23,27)(21,24,29,31)(26,28,30,32);(1,2,4,6)(3,5,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,26)(22,23)(24,28)(25,27)(29,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3,4,7)(2,5,6,8)(17,21,22,29)(18,24,25,31)(19,26,23,30)(20,28,27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=f,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 | #7232339, #7232343, #7232354, … |
| maximal normal subgroup of | #7232339, #7232343, #7232354, … |
| maximal quotient of | #582228, #582439, #582440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46101);