Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,32,29)(27,36,28,39)(30,34,31,35)(33,38,40,37);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,37,32,28,31,38)(26,33,35,36,29,40,34,39);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,31,37,32,27,30,38)(26,33,34,39,29,40,35,36);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,32,26)(27,33,28,40)(30,34,31,35)(36,38,39,37);(12,19)(16,22)(18,23)(20,24)(25,30,32,31)(26,34,29,35)(27,37,28,38)(33,39,40,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475966);