Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1438 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26,30,31)(27,33,35,29)(28,32,36,39)(34,40,37,38);(1,3)(2,6)(4,7)(5,8)(11,15)(14,16)(17,19,22,20)(18,23,21,24)(26,31)(28,36)(29,33)(34,37);(1,3,5,8)(2,7,4,6)(9,10,12,13)(11,14,15,16)(17,20,22,19)(18,23,21,24)(25,27)(26,29)(28,37)(30,35)(31,33)(32,40)(34,36)(38,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,23,20,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(1,4,5,2)(3,6,8,7)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419185);