Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,20,22,12)(9,13)(10,14)(11,17,26,35)(16,24)(18,31)(19,32)(21,29,36,27)(23,30)(25,34,28,33);(2,4)(5,7,10,19)(6,11,14,26)(8,17,22,35)(9,16,23,18)(12,15,20,32)(13,28,30,25)(21,33,36,34)(24,29,31,27);(6,12)(7,16)(9,23)(11,24)(13,29)(14,20)(15,28)(17,33)(18,19)(21,36)(25,32)(26,31)(27,30)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gh,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54919);