Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,29)(22,32)(23,39)(25,33)(27,34)(28,36)(30,35)(31,38)(37,40);(9,11,13,24,15,22,14,25)(10,16,17,21,20,33,19,32)(12,27,28,38,30,40,29,23)(18,34,35,39,36,37,26,31);(1,2,3,4)(10,17)(11,22)(13,14)(16,21)(18,26)(19,20)(23,38)(28,29)(31,37)(32,33)(34,39)(35,36);(6,8)(9,12,15,30)(10,18,20,36)(11,23,22,38)(13,29,14,28)(16,31,33,39)(17,26,19,35)(21,37,32,34)(24,40,25,27)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=i,e^2=i,f^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,g^b=gi,e^c=ei,f^c=fi,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, 330470);