Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #165149 |
| 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 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,7)(9,10,13,18,14,17,11,15)(12,20,19,24,22,16,21,23)(25,26)(27,34,36,39)(28,37)(29,32)(30,33)(31,35,40,38);(1,2,4,5)(3,7,8,6)(9,11,14,13)(10,15,17,18)(12,21,22,19)(16,24,20,23);(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,3,4,8)(2,6,5,7)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=h,d^2=ei,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,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, 131395);