Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,5)(6,7)(9,10,13,15)(11,12,16,14)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,28,35,32)(29,33,37,39)(34,36,38,40);(1,3)(2,4)(5,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19,21,18)(20,23,24,22)(26,31)(28,32)(29,37)(34,38);(2,7)(3,5)(9,10,13,15)(11,12,16,14)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,37)(39,40);(1,4,6,8)(2,3,7,5)(9,10)(11,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,37,40)(33,34,39,38);(1,5,6,3)(2,8,7,4)(9,12)(10,11)(13,14)(15,16)(25,29)(26,33)(27,34)(28,36)(30,37)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4980992);