Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,15,25,12)(7,16,19,11,20,27,17,28)(8,21,23,13,24,30,33,29)(18,26,34,36,22,35,32,31);(1,2)(3,4)(5,7,9,19,10,20,25,17)(6,11,12,16,15,28,14,27)(8,22,23,32,24,18,33,34)(13,26,21,31,29,35,30,36);(2,4)(6,12)(7,17)(8,23)(9,25)(13,29)(14,15)(16,27)(19,20)(24,33)(26,31)(32,34)(35,36);(5,8,9,23,10,24,25,33)(6,13,12,21,15,29,14,30)(7,18,19,34,20,22,17,32)(11,26,16,31,28,35,27,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383897);