Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| 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 | |
1378 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)(5,6,11,17,9,15,28,21,13,18,10,16,12,7,19,22)(8,23,26,20,31,36,30,34,27,33,25,32,24,35,29,14);(1,2,3,4)(5,7,13,15)(6,9,18,12)(8,20,27,32)(10,17,11,16)(14,24,34,31)(19,22,28,21)(23,25,33,26)(29,36,30,35);(5,8)(6,14)(7,20)(9,24)(10,29)(11,30)(12,31)(13,27)(15,32)(16,35)(17,36)(18,34)(19,25)(21,23)(22,33)(26,28);(5,9,13,12)(6,15,18,7)(8,24,27,31)(10,19,11,28)(14,32,34,20)(16,22,17,21)(23,35,33,36)(25,30,26,29)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226791, #72226794, #72226796, … |
| maximal normal subgroup of | #72226791, #72226794, #72226796, … |
| maximal quotient of | #72285656, #72310178, #72319637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401732);