Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 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)(5,6,9,14,23,20,10,30,12,16,29,19,11,7,17,15)(8,21,25,18,31,32,36,34,28,22,24,33,27,35,26,13);(2,4)(5,7)(6,11)(8,22,28,21)(9,19)(10,30)(12,20)(13,25,34,24)(14,29)(15,17)(16,23)(18,26,33,36)(27,35,31,32);(2,4)(5,8,23,27,12,28,11,31)(6,13,20,33,16,34,7,18)(9,26,10,24,29,36,17,25)(14,35,30,22,19,32,15,21)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56588);