Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(14,19)(15,23)(16,27)(17,30)(20,37)(21,41)(22,36)(24,40)(25,38)(26,33)(28,32)(31,34)(35,39)(42,43);(2,4)(13,14,18,22)(15,24,26,37)(16,28,31,41)(17,32,30,21)(19,34,36,27)(20,38,40,25)(23,43,33,35)(29,42,44,39);(6,8)(13,15,18,26)(14,20,22,40)(16,29,31,44)(17,33,30,23)(19,35,36,43)(21,42,32,39)(24,41,37,28)(25,34,38,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55106);