Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,9)(10,23)(13,31)(14,30)(15,16)(18,32)(19,28)(20,25)(21,35)(22,27)(24,26)(29,34);(1,2)(3,4)(5,6)(7,18)(8,22)(9,12)(10,27)(11,14)(13,20)(15,24)(16,23)(17,29)(19,36)(21,31)(25,33)(26,30)(28,32)(34,35);(2,4)(5,7)(6,13)(8,17)(9,25)(10,28)(11,21)(12,29)(14,32)(15,33)(16,31)(18,27)(19,26)(20,23)(22,36)(24,35)(30,34);(6,14)(12,30)(13,32)(15,22)(16,27)(18,31)(29,34)(33,36)Polycyclic presentation:
2^9|e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72224952, #72226813, … |
| maximal normal subgroup of | #72224928, #72224952, #72226813, … |
| maximal quotient of | #72224922 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406063);