Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(10,13)(11,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,30)(31,32);(1,2,6,7)(3,4,5,8)(10,13)(14,16)(18,23)(19,26)(20,29)(30,31);(1,3,6,5)(2,8,7,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,24,23,21)(20,27,29,32)(25,31,28,30);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32);(1,5,6,3)(2,8,7,4)(17,21,22,24)(18,26,23,19)(20,30,29,31)(25,32,28,27)Polycyclic presentation:
2^8|b^2=fgh,c^2=f,d^2=g,e^2=f,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 | C2×#52773, #8092525, #8092679, … |
| maximal normal subgroup of | C2×#52773, #8092525, #8092679, … |
| maximal quotient of | #582348, #582517, #582548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45761);