Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 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)(7,12,19,28)(8,20,24,29)(9,14)(10,16)(11,15)(13,23,30,22)(17,32)(18,27,25,31)(21,34)(26,35)(33,36);(1,2)(3,4)(5,7,10,25,11,19,9,18)(6,12,14,27,15,28,16,31)(8,21,23,33,24,17,22,35)(13,26,29,32,30,36,20,34);(2,4)(5,8)(6,13,15,30)(7,17,19,21)(9,22)(10,23)(11,24)(12,26)(14,29,16,20)(18,33,25,35)(27,32)(28,36)(31,34);(5,9,11,10)(6,14,15,16)(7,18,19,25)(8,22,24,23)(12,27,28,31)(13,29,30,20)(17,33,21,35)(26,32,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,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, 265529);