Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16,12,17,27,32)(7,18,21,23,13,30,31,36)(8,22,24,19,25,20,35,26)(9,14,10,15,28,33,29,34);(2,4)(5,7,12,13)(6,8,17,25)(9,26,28,19)(10,22,29,20)(11,21,27,31)(14,30,33,18)(15,23,34,36)(16,35,32,24);(5,8)(6,13)(7,17)(9,18)(10,23)(11,24)(12,25)(14,26)(15,22)(16,31)(19,33)(20,34)(21,32)(27,35)(28,30)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54959);