Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #367 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,20,21,36)(15,25)(16,30)(17,33)(18,27)(22,41,40,23)(24,44,37,39)(26,34)(28,32)(29,31)(35,42,43,38);(2,4)(13,14,16,22,17,23,15,20,27,37,25,38,26,35,30,44,19,21,32,40,34,41,29,36,18,24,31,42,33,43,28,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286481, #72286690, #72286713, … |
| maximal normal subgroup of | #72286481, #72286690, #72286713, … |
| maximal quotient of | #72286496, #72321153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1219);