Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(10,16)(11,20)(12,24)(13,14)(17,31)(19,28)(23,29)(25,30)(26,27)(32,39)(35,37)(36,40);(1,2,5,6)(3,7,4,8)(9,10,15,19)(11,21,23,33)(12,25,27,30)(13,28,14,16)(17,29,31,20)(18,24,34,26)(22,32,38,40)(35,39,37,36);(1,3,5,4)(2,7,6,8)(9,11,14,29,15,23,13,20)(10,17,28,33,19,31,16,21)(12,22,26,37,27,38,24,35)(18,32,25,39,34,40,30,36);(1,4,5,3)(2,7,6,8)(9,12,15,27)(10,18,19,34)(11,22,23,38)(13,24,14,26)(16,30,28,25)(17,32,31,40)(20,35,29,37)(21,36,33,39)Polycyclic presentation:
2^9|b^2=ef,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351729);