Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| 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 | |
1362 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)(5,6,12,18,9,16,10,28,14,19,11,22,13,7,20,17)(8,23,26,31,30,36,29,15,27,33,35,21,24,32,25,34);(1,2,3,4)(5,7,13,6,14,16,9,19)(8,21,30,15,27,31,24,34)(10,28,12,22,20,17,11,18)(23,25,32,26,33,29,36,35);(5,8)(6,15)(7,21)(9,24)(10,26)(11,25)(12,29)(13,30)(14,27)(16,31)(17,23)(18,32)(19,34)(20,35)(22,36)(28,33);(5,9,14,13)(6,16,19,7)(8,24,27,30)(10,11,20,12)(15,31,34,21)(17,18,28,22)(23,32,33,36)(25,35,29,26)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,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 | #72225127, #72249871, #72249902, … |
| maximal normal subgroup of | #72225127, #72249871, #72249902, … |
| maximal quotient of | #72225100, #72225109, #72286382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401764);