Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,15,11,17,9,23)(7,18,21,29,20,34,8,13)(10,16,24,30,25,33,26,27)(14,19,32,22,28,35,31,36);(1,2)(3,4)(5,7,9,8,11,20,12,21)(6,13,15,18,17,29,23,34)(10,19,24,22,25,35,26,36)(14,30,31,16,28,27,32,33);(2,4)(5,8,9,20,11,21,12,7)(6,14,15,31,17,28,23,32)(10,22,24,35,25,36,26,19)(13,27,18,33,29,30,34,16)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^c=ghi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55896);