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)(7,17,21,29)(8,10,14,9)(11,15,28,33)(12,16)(13,19)(18,26)(20,35)(22,23)(24,31)(25,27)(30,34,32,36);(5,7,12,21)(6,13,16,31)(8,17,14,29)(9,19,10,24)(11,26,28,20)(15,32,33,30)(18,27,35,22)(23,34,25,36);(2,4)(5,8,9,6)(7,18,19,32)(10,16,12,14)(11,27,23,33)(13,26,17,34)(15,28,22,25)(20,29,36,31)(21,35,24,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=ehi,f^c=fi,g^c=ghi,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, 55668);