Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,19,14,16,11,18,13,20,10,17,9,7,21,23)(8,24,28,15,25,34,31,22,30,36,26,35,29,33,27,32);(1,2,3,4)(5,7,13,16)(6,14,20,9)(8,22,30,32)(10,19,12,17)(11,18,21,23)(15,29,35,25)(24,27,36,31)(26,33,28,34);(5,8)(6,15)(7,22)(9,25)(10,27)(11,28)(12,31)(13,30)(14,29)(16,32)(17,24)(18,34)(19,36)(20,35)(21,26)(23,33);(5,9,13,14)(6,16,20,7)(8,25,30,29)(10,11,12,21)(15,32,35,22)(17,23,19,18)(24,33,36,34)(26,27,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250063, #72250068, #72250079, … |
| maximal normal subgroup of | #72250063, #72250068, #72250079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401724);