Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,26,20,27,17,24,21,28,42,36,41,30,18,25)(15,29,33,22,23,44,32,37,34,16,35,39,43,40,31,38);(2,4)(5,7,9,8)(6,10,11,12)(13,15,21,34)(14,22,28,39)(16,36,29,26)(17,33,18,35)(19,32,42,31)(20,43,41,23)(24,44,25,40)(27,38,30,37);(2,4)(5,8,9,7)(6,10,11,12)(13,16,21,29)(14,23,28,43)(15,30,34,27)(17,39,18,22)(19,38,42,37)(20,44,41,40)(24,32,25,31)(26,35,36,33)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56599);