Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1775 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(1,2,5,6)(3,8,7,4)(9,10,12,14)(11,15,16,13)(18,20)(19,24)(27,34)(28,33)(35,40)(36,39);(2,6)(3,7)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,23,24,22)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,23,24,22)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=ghi,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1545062);