Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10)(11,15,21,30)(12,23,24,32)(13,19)(14,18)(16,25,33,38)(17,26)(20,35)(22,29,27,34)(28,39)(31,37)(36,40);(1,2,4,5)(3,8,7,6)(9,11,13,21)(10,15,19,30)(12,20,24,36)(14,27,26,22)(16,31,33,28)(17,29,18,34)(23,35,32,40)(25,39,38,37);(9,12)(10,16,19,33)(11,20)(13,24)(14,25)(15,28,30,31)(17,32,18,23)(21,36)(22,37)(26,38)(27,39)(29,35,34,40);(1,3,4,7)(2,6,5,8)(10,17,19,18)(11,21)(15,29,30,34)(16,32,33,23)(20,36)(22,27)(28,35,31,40)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294322);