Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| 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 | |
1005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,14,17,15,19,13,20)(11,21,24,31,25,16,22,36)(12,18,28,34,29,35,27,32)(23,30,38,33,39,37,26,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,24)(14,22)(15,25)(17,31)(18,33)(19,21)(20,36)(23,27)(28,39)(29,38)(30,34)(32,37)(35,40);(1,3)(2,6)(4,8)(5,7)(10,17)(11,22)(13,14)(18,34)(19,20)(23,38)(24,25)(26,39)(27,28)(30,37)(31,36)(32,35);(3,8)(5,6)(9,12,15,29)(10,18,19,35)(11,23,25,39)(13,28,14,27)(16,30,21,37)(17,32,20,34)(22,38,24,26)(31,33,36,40)Polycyclic presentation:
2^9|a^2=eg,d^2=i,f^2=i,g^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,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385477);