Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1694 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,24,20)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,39)(36,40);(2,3)(4,8)(17,18)(19,23)(20,24)(21,22)(25,27,31,28)(26,32,30,34)(29,35,38,36)(33,39,37,40);(2,3)(6,7)(18,22)(19,24)(25,28,31,27)(26,32,30,34)(29,35,38,36)(33,40,37,39);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,20,22,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,23)(22,24)(25,30,31,26)(27,32,28,34)(29,33,38,37)(35,40,36,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3134832);