Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26)(27,34)(28,30)(29,33)(31,35)(32,37)(36,40)(38,39);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(26,30)(28,37)(31,34)(32,33)(35,38)(39,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24);(11,21)(15,18)(19,20)(23,24)(25,27,29,36)(26,31,33,39)(28,38,37,35)(30,34,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6600520);