Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4)(5,6,7,8)(9,10)(11,22)(12,28)(13,31)(14,20)(15,19)(16,21)(17,25)(18,32)(23,39)(24,35)(26,37)(27,36)(29,40)(30,33)(34,38);(2,4)(9,11,16,27)(10,17,21,37)(12,29,18,35)(13,24,33,40)(14,26,15,25)(19,36,20,22)(23,31,38,30)(28,34,32,39);(6,8)(9,12,16,18)(10,13,21,33)(11,23,27,38)(14,31,15,30)(17,34,37,39)(19,28,20,32)(22,24,36,40)(25,35,26,29);(9,13)(10,18)(11,24)(12,21)(14,28)(15,32)(16,33)(17,35)(19,30)(20,31)(22,38)(23,36)(25,39)(26,34)(27,40)(29,37)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302664);