Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1926 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,19,22,27)(8,21,28,35)(9,16)(10,31)(11,24)(12,18)(13,20,29,36)(14,25)(15,30)(17,32)(23,34,33,26);(1,3,4,2)(5,7)(6,13)(8,24)(9,21)(10,20)(11,33)(12,22)(14,34)(15,19)(16,27)(17,28)(18,29)(23,32)(25,36)(26,31)(30,35);(2,3)(5,8,25,27,32,13,9,26,12,28,10,19,11,29,30,34)(6,7,16,35,24,33,14,20,18,22,15,21,17,23,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56366);