Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6)(7,8)(9,10,15,19,13,21,14,20)(11,22,24,17,26,16,25,33)(12,27,29,18,31,36,30,35)(23,34,39,32,28,40,38,37);(1,2)(3,4)(9,11,15,25,13,26,14,24)(10,16,20,33,21,22,19,17)(12,28,30,38,31,23,29,39)(18,32,36,34,35,37,27,40);(2,4)(9,11)(10,17)(12,23)(13,26)(14,25)(15,24)(16,19)(18,34)(20,22)(21,33)(27,37)(28,31)(29,39)(30,38)(32,36)(35,40);(6,8)(9,12,15,29,13,31,14,30)(10,18,19,36,21,35,20,27)(11,23,24,39,26,28,25,38)(16,32,33,40,22,37,17,34)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,d^2=ef,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,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337650);