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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(18,22)(20,23)(25,26)(27,29)(28,36)(30,31)(32,35)(33,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,12,16)(18,22)(19,24)(26,31)(28,37)(29,32)(34,38);(2,7)(3,8)(9,10,14,12)(11,16,13,15)(17,18,21,22)(19,23,24,20)(25,27)(26,32)(28,38)(29,31)(30,35)(33,39)(34,37)(36,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,33)(27,34)(29,39)(30,37)(31,36)(32,40)(35,38);(1,4)(2,3)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(17,20,21,23)(18,19,22,24)(25,29,30,32)(26,27,31,35)(28,39,37,40)(33,34,36,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=fg,d^2=fi,e^2=fh,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, 4419243);