Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2456 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)(3,4,7,8)(5,6)(9,10,14,19,15,20,13,18)(11,21,23,16,25,30,24,31)(12,17,27,33,28,34,26,32)(22,36,37,40,38,35,39,29);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,29)(18,30)(19,21)(20,31)(26,37)(27,39)(28,38)(32,36)(33,35)(34,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,29)(18,32)(19,33)(20,34)(21,35)(23,37)(24,39)(25,38)(30,36)(31,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,26,28,27)(16,30,31,21)(17,32,34,33)(22,37,38,39)(29,36,40,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94493);