Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,12)(9,24)(10,27)(11,14)(13,19)(15,22)(16,25)(17,23)(20,35)(21,31)(26,29)(28,33)(30,36)(32,34);(6,12)(7,19)(8,22)(9,25)(10,23)(14,24)(15,17)(16,27)(18,28)(20,36)(21,26)(30,32)(31,35)(33,34);(2,4)(5,7,8,19,23,33,25,36,11,21,9,26,10,20,22,34)(6,13,15,18,27,32,24,35,14,29,16,31,17,30,12,28);(6,14)(12,24)(13,29)(15,16)(17,27)(18,31)(28,35)(30,32)Polycyclic presentation:
2^9|c^2=e,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239804, #72239809, #72239838, … |
| maximal normal subgroup of | #72239804, #72239809, #72239838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406355);