Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10980 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1076 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)(3,14,18,30)(4,10,19,8)(5,11,21,24)(6,12)(7,13)(9,17)(15,25,22,23)(16,26)(20,28)(27,31)(29,32);(2,8)(3,15)(4,19)(6,20)(9,14)(10,13)(11,12)(16,17)(18,22)(23,26)(24,28)(25,29)(27,30)(31,32);(1,3)(2,9)(4,16)(5,17)(6,22)(7,18)(8,23)(10,25)(11,14)(12,29)(13,27)(15,20)(19,32)(21,31)(24,30)(26,28)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,f^c=fi,g^c=gh,g^d=gi,g^e=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54941);