Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147614 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10966 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,25)(4,16,19,8)(5,10)(6,20)(7,11)(9,13,26,21)(14,24)(17,27,18,23)(22,30)(28,32)(29,31);(2,8)(3,13)(5,6)(9,24)(10,27)(11,16)(12,28)(14,30)(15,21)(17,18)(20,23)(22,25)(26,29)(31,32);(1,3)(2,9,11,26)(4,14)(5,13)(6,21)(7,15)(8,22,16,28)(10,12,20,25)(17,30)(18,32)(19,31)(23,29,27,24)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^c=gi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54810);