Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,20,24,34)(9,25)(10,14)(11,15)(12,19)(13,23,31,22)(17,27,21,30)(18,29)(26,28)(32,33,36,35);(1,2)(3,4)(5,7,10,19,11,18,9,26)(6,12,14,29,15,28,25,16)(8,21,23,35,24,17,22,33)(13,30,20,32,31,27,34,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,20,15,31,25,34)(7,17,19,33,18,21,26,35)(12,27,16,32,28,30,29,36);(7,18)(12,28)(16,29)(17,21)(19,26)(27,30)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265472);