Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #390 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,23,26,19)(16,27,30,20)(17,22)(18,21)(24,34)(25,39,31,35)(28,36)(29,40,32,37)(33,38)(41,42)(43,44);(2,4)(13,15,18,26)(14,19,21,23)(16,28,30,43)(17,31,33,25)(20,36,27,44)(22,39,38,35)(24,32,42,29)(34,40,41,37);(13,16,18,30)(14,20,21,27)(15,24,26,42)(17,32,33,29)(19,34,23,41)(22,40,38,37)(25,43,31,28)(35,44,39,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18419);