Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2305 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)(9,10,14,19,15,20,13,18)(11,21,23,32,25,35,24,16)(12,26,29,34,30,36,28,17)(22,31,38,33,39,40,27,37);(1,2,3,4)(9,11,14,23,15,25,13,24)(10,16,18,35,20,32,19,21)(12,27,29,22,30,38,28,39)(17,33,36,31,34,37,26,40);(6,8)(9,12,15,30)(10,17,20,34)(11,22,25,39)(13,29,14,28)(16,31,32,40)(18,26,19,36)(21,37,35,33)(23,27,24,38);(2,4)(10,18)(11,23)(13,14)(16,32)(17,26)(19,20)(22,27)(24,25)(28,29)(31,40)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,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, 326188);