Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476654 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,15,17,14,20,13,19)(11,21,29,16,22,31,24,33)(12,25,27,18,28,34,30,36)(23,35,39,37,26,32,38,40);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,16,20,33)(12,26,28,23)(13,29,15,24)(17,21,19,31)(18,35,36,32)(25,37,34,40)(27,39,30,38);(1,4,5,3)(2,6,7,8)(10,17)(11,22)(12,27)(13,15)(16,31)(18,36)(19,20)(21,33)(23,38)(26,39)(28,30)(32,35);(3,4)(6,8)(9,12)(10,18)(11,23)(13,30)(14,28)(15,27)(16,32)(17,34)(19,25)(20,36)(21,37)(22,26)(24,38)(29,39)(31,40)(33,35)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386111);