Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,13,24,29,11,15,25,31,8,21,22,30)(7,16,19,33,20,27,34,36,17,32,23,35,18,12,26,28);(1,2,3,4)(5,7,8,18,11,17,10,20)(6,12,13,16,15,27,21,32)(9,23,22,34,25,19,24,26)(14,28,29,33,31,36,30,35);(7,17)(12,27)(16,32)(18,20)(19,23)(26,34)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34122);