Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164512 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22)(25,26)(27,31)(28,35)(29,37)(30,33)(32,34)(36,39)(38,40);(1,2,5,6)(3,7,4,8)(25,27,30,29)(26,31,33,37)(28,34,36,38)(32,39,40,35);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,28,30,36)(26,32,33,40)(27,34,29,38)(31,39,37,35);(1,4,5,3)(2,7,6,8)(26,33)(31,37)(32,40)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=ei,c^2=eh,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,f^c=fi,g^c=ghi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132130);