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 | #2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
854 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)(10,11)(14,15)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31);(2,3)(4,8)(9,10,13,11)(12,15,16,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^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 | #7206225, #7206229, #7206233, … |
| maximal normal subgroup of | #7206225, #7206229, #7206233, … |
| maximal quotient of | #582156, #582367, #582369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46010);