Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501527 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,11,17,18,15,20,8,13,16,19,10,12,7,14);(1,2)(3,4)(7,15)(9,16)(11,19)(14,20);(2,4)(5,6)(7,11)(8,13)(9,14)(10,18)(12,17)(15,19)(16,20);(5,7,10,16,8,15,17,9)(6,11,18,20,13,19,12,14)Polycyclic presentation:
2^9|a^2=f,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72224853 |
| maximal normal subgroup of | #72224853 |
| maximal quotient of | #72224849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408724);