Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,21,25,33,23,32,24,16)(12,26,29,36,30,35,28,17)(22,37,27,40,38,31,39,34);(2,5)(6,8)(9,11,15,23)(10,16)(12,27)(13,24,14,25)(17,34,36,40)(18,32)(19,21)(20,33)(22,28)(26,37,35,31)(29,38)(30,39);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,30,14,29)(10,17,19,26,20,36,18,35)(11,22,24,39,23,38,25,27)(16,31,21,34,33,37,32,40);(2,5)(6,8)(10,18,20,19)(11,23)(16,21,33,32)(17,35,36,26)(22,38)(24,25)(27,39)(31,34,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286468);