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 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,23,24,21)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,2,5,4)(3,6,7,8)(10,15)(11,18)(17,20)(19,24)(27,35)(29,39)(31,40)(33,36);(1,3)(2,6)(4,8)(5,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4,5,2)(3,8,7,6)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=g,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×#41658, #7639866, #7640239, … |
| maximal normal subgroup of | C2×#41658, #7639866, #7640239, … |
| maximal quotient of | #544797, #545247, #545861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34646);