Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,7,9,13,16,12)(8,15,17,20,18,14,19,11);(5,7,10,13,9,12,16,6)(8,15,17,20,18,14,19,11);(5,8,16,17,9,18,10,19)(6,11,12,15,13,20,7,14);(2,4)(6,12)(7,13)(10,16)(11,15)(14,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385032);