Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,15)(11,12,16,14)(19,23)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(18,21)(22,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(2,6)(3,7)(9,11,13,16)(10,14,15,12)(17,18,20,21)(19,22,23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72267448, #72267455, #72267626, … |
| maximal normal subgroup of | #72267448, #72267455, #72267626, … |
| maximal quotient of | #72267470, #72267675, #72267725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4870982);