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 | |
545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,19)(11,20,24,36)(12,25)(13,17,30,18)(15,22,32,23)(16,29)(21,37)(26,40)(27,35)(28,34)(31,38)(33,39);(1,3,5,8)(2,6,4,7)(9,11,14,24)(10,15,19,32)(12,26,29,38)(13,22,30,23)(16,33,25,37)(17,20,18,36)(21,27,39,28)(31,35,40,34);(1,4,5,2)(3,7,8,6)(9,12,13,27,14,29,30,28)(10,16,17,34,19,25,18,35)(11,21,22,26,24,39,23,38)(15,31,20,37,32,40,36,33);(3,8)(6,7)(10,17,19,18)(11,22,24,23)(16,34,25,35)(21,26,39,38)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361049);