Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(9,10,13,20,15,18,14,19)(11,21,24,16,25,31,23,32)(12,26,28,17,29,34,27,33)(22,30,36,35,38,40,37,39);(1,2)(3,4)(9,11,14,23,15,25,13,24)(10,16,20,31,18,32,19,21)(12,22,28,36,29,38,27,37)(17,30,26,39,33,40,34,35);(2,4)(5,7,8,6)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,34)(20,26)(21,35)(23,36)(24,37)(25,38)(31,39)(32,40);(10,18)(11,23)(13,14)(16,21)(17,33)(22,36)(24,25)(27,28)(30,35)(31,32)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296318);