Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,13,15)(11,17,18,24)(12,20,22,16)(14,21,23,19)(25,26,30,33,31,28,29,38)(27,32,36,39,34,37,35,40);(1,2,5,6)(3,7,4,8)(11,18)(14,23)(17,24)(19,21)(25,27,29,35,31,34,30,36)(26,32,38,40,28,37,33,39);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,14,15,23)(12,21,22,19)(16,24,20,17)(25,28,31,26)(27,32,34,37)(29,33,30,38)(35,40,36,39);(1,4,5,3)(2,7,6,8)(10,15)(14,23)(16,20)(17,24)(26,28)(27,34)(33,38)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=eh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=dei,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138176);