Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
762 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)(7,16,21,32)(8,22,13,9)(10,25,28,14)(11,15)(12,18)(17,29)(19,31)(20,35)(23,27)(24,26)(30,36,34,33);(5,7,11,21)(6,12,15,31)(8,16,13,32)(9,19,22,18)(10,20,28,29)(14,30,25,34)(17,26,35,23)(24,33,27,36);(2,4)(5,8,9,6,11,13,22,15)(7,17,18,34,21,35,19,30)(10,26,24,25,28,23,27,14)(12,29,16,33,31,20,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55671);