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 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26,30,29)(27,32,35,31)(28,33,38,37)(34,40,36,39);(2,3)(4,8)(9,11)(10,13)(12,14)(15,16)(18,21)(19,24)(25,27,30,35)(26,31,29,32)(28,34,38,36)(33,39,37,40);(2,3)(6,7)(10,15)(11,12)(17,18,20,21)(19,22,24,23)(25,27)(26,32)(28,36)(29,31)(30,35)(33,39)(34,38)(37,40);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,15,13)(17,18)(19,22)(20,21)(23,24)(25,28)(26,33)(27,34)(29,37)(30,38)(31,39)(32,40)(35,36);(1,4,5,8)(2,6,3,7)(9,13,14,16)(10,11,15,12)(17,19,20,24)(18,22,21,23)(25,29,30,26)(27,31,35,32)(28,37,38,33)(34,39,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fghi,d^2=fhi,e^2=f,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, 1625272);