Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1938 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)(18,20)(19,23)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,39)(38,40);(11,16)(13,15)(18,20)(22,24)(26,27)(28,35)(29,37)(39,40);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,39,38,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,22,20,24)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028741);