Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | #19619 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6,9,13)(8,10,15,16)(17,18,21,26,22,27,31,25)(19,23,20,29,28,24,30,32);(1,2,7,6,5,9,4,13)(3,10,14,15,11,16,12,8)(17,19,21,20,22,28,31,30)(18,23,26,29,27,24,25,32);(3,11)(8,15)(10,16)(12,14)(17,20,21,28,22,30,31,19)(18,24,25,29,27,23,26,32);(1,3,4,12,5,11,7,14)(2,8,6,10,9,15,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322005, #72322026, #72322204 |
| maximal normal subgroup of | #72322005, #72322026, #72322204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138929);