Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,16,8,14,10,15)(7,17,21,30,19,12,18,27)(9,13,23,29,22,28,24,31)(20,33,34,32,35,26,36,25);(1,2,4,3)(5,7,8,19)(6,12,14,17)(9,20,22,35)(10,18,11,21)(13,26,28,33)(15,30,16,27)(23,34,24,36)(25,31,32,29);(1,3,4,2)(6,13)(7,18)(10,11)(12,25)(14,28)(15,31)(16,29)(17,32)(19,21)(20,34)(23,24)(26,30)(27,33)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=di,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gh,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55894);