Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| 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:
(1,2,5,8)(3,6,4,7)(10,12)(11,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,31)(27,30)(29,32);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,16,15,14)(18,23)(21,24)(25,28)(30,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,15)(10,14,12,16)(17,19,22,26)(18,24,23,21)(20,29,31,27)(25,30,28,32);(9,12,13,10)(11,16,15,14)(17,20,22,31)(18,25,23,28)(19,27,26,29)(21,30,24,32);(1,4,5,3)(2,7,8,6)(17,21,22,24)(18,26,23,19)(20,30,31,32)(25,29,28,27)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41684, #7640116, #7640240, … |
| maximal normal subgroup of | C2×#41684, #7640116, #7640240, … |
| maximal quotient of | #544749, #545236, #545791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34672);