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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(10,14)(12,15)(19,23)(21,24)(25,26,31,29)(27,33,28,32)(30,37,38,34)(35,39,36,40);(3,7)(6,8)(10,14)(11,16)(25,27,31,28)(26,32,29,33)(30,36,38,35)(34,40,37,39);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,22)(19,21,23,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,3,5,7)(2,6,4,8)(9,11,13,16)(10,15,14,12)(17,19,20,23)(18,21,22,24)(25,29,31,26)(27,32,28,33)(30,37,38,34)(35,39,36,40);(1,4,5,2)(3,8,7,6)(9,12,13,15)(10,16,14,11)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,d^2=fhi,e^2=g,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, 1460352);