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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,3,5,4)(2,6,8,7)(10,14)(15,17)(16,20)(23,24)(27,35)(29,39)(31,40)(33,36);(1,2)(3,6)(4,7)(5,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(1,3,5,4)(2,6,8,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=g,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×#41855, #7640048, #7640248, … |
| maximal normal subgroup of | C2×#41855, #7640048, #7640248, … |
| maximal quotient of | #544959, #545331, #545976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34843);