Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
979 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)(7,8)(9,10)(11,26)(13,30)(14,31)(15,16)(17,35)(19,22)(20,23)(21,25)(24,28)(27,29)(32,36)(33,34);(5,6)(7,13)(8,14)(9,24)(10,28)(11,17)(12,18)(15,27)(16,29)(19,31)(20,34)(21,33)(22,30)(23,36)(25,32)(26,35);(2,4)(5,7,11,23,12,19,26,21)(6,13,17,36,18,31,35,33)(8,9,25,10,22,27,20,29)(14,24,32,28,30,15,34,16);(5,8)(6,14)(7,9)(10,23)(11,25)(12,22)(13,24)(15,31)(16,33)(17,32)(18,30)(19,27)(20,26)(21,29)(28,36)(34,35)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=ghi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250062, #72250067, #72250074, … |
| maximal normal subgroup of | #72250062, #72250067, #72250074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419308);