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 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,21)(9,14)(10,30)(11,33)(12,18)(13,25)(15,28)(16,32)(17,29)(20,31)(22,24)(23,36)(26,35)(27,34);(1,2)(3,4)(5,7)(6,8)(9,13)(10,22)(11,34)(12,23)(14,26)(15,25)(16,36)(17,24)(18,27)(19,30)(20,29)(21,28)(31,33)(32,35);(2,4)(5,8)(6,13)(7,14)(9,24)(10,31)(11,19)(12,27)(15,34)(16,35)(17,23)(18,20)(21,33)(22,30)(25,32)(26,29)(28,36);(5,9,12,29)(6,14,18,17)(7,20,23,13)(8,24,27,26)(10,28,32,11)(15,16,33,30)(19,31,36,25)(21,22,34,35)Polycyclic presentation:
2^9|d^2=i,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, #72224933, #72226819, … |
| maximal normal subgroup of | #72224928, #72224933, #72226819, … |
| maximal quotient of | #72224923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406065);