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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
722 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)(3,5)(4,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31);(10,11)(12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(2,3)(6,7)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)Polycyclic presentation:
2^8|a^2=h,c^2=f,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 | #7221604, #7221608, #7221668, … |
| maximal normal subgroup of | #7221604, #7221608, #7221668, … |
| maximal quotient of | #582292, #582493, #582534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46026);