Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,15,16,13)(25,26,30,31)(27,33,35,29)(28,36,37,32)(34,40,38,39);(11,16)(13,15)(17,18,21,20)(19,23,24,22)(26,31)(28,37)(29,33)(34,38);(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,29)(28,38)(30,35)(31,33)(32,39)(34,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,20)(19,22,24,23)(25,28,30,37)(26,32,31,36)(27,34,35,38)(29,40,33,39);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,29)(26,27)(28,39)(30,33)(31,35)(32,38)(34,36)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=fh,e^2=ghi,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, 3140813);