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 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 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)(3,7)(4,6)(5,8)(10,14)(11,17)(12,20)(22,24)(25,26,30,31)(27,32,33,28)(29,37,38,34)(35,36,40,39);(1,3,6,5)(2,7,4,8)(26,31)(28,32)(34,37)(36,39);(1,4,6,2)(3,7,5,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,5,6,3)(2,7,4,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=fg,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×#52810, #8092666, #8092702, … |
| maximal normal subgroup of | C2×#52810, #8092666, #8092702, … |
| maximal quotient of | #582330, #582516, #582543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45798);