Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,18,15,19,14,29)(11,20,21,33,23,16,24,31)(12,25,27,17,28,35,30,36)(22,38,37,34,26,40,39,32);(1,3,4,8)(2,5,6,7)(9,11)(10,16,19,20)(12,26,28,22)(13,21)(14,24)(15,23)(17,34)(18,31,29,33)(25,38)(27,39,30,37)(32,36)(35,40);(2,6)(5,7)(11,21,23,24)(16,31,20,33)(22,37,26,39)(32,38,34,40);(3,8)(5,7)(9,12,13,27,15,28,14,30)(10,17,18,35,19,36,29,25)(11,22,21,37,23,26,24,39)(16,32,31,38,20,34,33,40)Polycyclic presentation:
2^9|a^2=eg,b^2=egh,c^2=ef,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361129);