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)(9,10,13,19,15,20,14,17)(11,21,22,16,25,33,24,31)(12,26,28,18,30,34,29,36)(23,32,27,37,39,40,38,35);(1,2,3,4)(9,11,13,22,15,25,14,24)(10,16,17,21,20,31,19,33)(12,27,28,39,30,38,29,23)(18,35,26,40,36,37,34,32);(6,8)(10,17)(11,22)(13,14)(16,31)(18,34)(19,20)(23,38)(24,25)(26,36)(27,39)(28,29)(32,40);(2,4)(9,12,15,30)(10,18,20,36)(11,23,25,39)(13,29,14,28)(16,32,31,40)(17,34,19,26)(21,37,33,35)(22,38,24,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^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, 326183);