Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2068 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(18,20)(19,23)(25,26,29,30)(27,33,35,31)(28,32,36,38)(34,40,37,39);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(18,20)(22,24)(26,30)(28,36)(31,33)(34,37);(2,3)(6,7)(10,11)(14,15)(17,18,21,20)(19,24,23,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=fghi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5692808);