Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,5)(17,18,21,19)(20,23,24,22)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(1,2)(3,7)(4,6)(5,8)(10,14)(15,16)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(1,3,6,5)(2,7,4,8)(9,10,13,14)(11,15,12,16)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(1,4,6,2)(3,7,5,8)(9,11,13,12)(10,15,14,16)(17,20,21,24)(18,22,19,23)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=hi,e^2=fh,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, 1758910);