Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1540 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)(9,10,16,13)(11,21,25,34)(12,26,31,36)(14,20,15,19)(17,24,32,23)(18,29,28,30)(22,37,39,33)(27,35,38,40);(5,6)(7,8)(9,11,14,23,16,25,15,24)(10,17,19,21,13,32,20,34)(12,27,29,22,31,38,30,39)(18,35,36,33,28,40,26,37);(1,2,3,4)(6,8)(9,12)(10,18)(11,22)(13,28)(14,30)(15,29)(16,31)(17,33)(19,26)(20,36)(21,35)(23,27)(24,38)(25,39)(32,37)(34,40);(9,13,16,10)(11,21,25,34)(12,28,31,18)(14,19,15,20)(17,24,32,23)(22,35,39,40)(26,29,36,30)(27,33,38,37)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330407);