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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
696 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(26,28)(27,34)(32,36)(35,40);(2,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=g,e^2=g,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×#52700, #8092148, #8092474, … |
| maximal normal subgroup of | C2×#52700, #8092148, #8092474, … |
| maximal quotient of | #582324, #582523, #582555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45688);