Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
559 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)(7,17)(8,20)(9,24)(10,27)(11,16)(12,18)(13,22)(14,26)(15,21)(19,32)(23,30)(25,33)(28,34)(29,36)(31,35);(1,2)(3,4)(5,7)(6,12)(8,21)(9,22)(10,23)(11,19)(13,15)(14,30)(16,28)(17,24)(18,26)(20,31)(25,36)(27,29)(32,33)(34,35);(2,4)(9,25)(14,31)(17,32)(18,34)(20,30)(22,36)(24,33)(26,35);(5,8,10,7,11,23,21,19)(6,13,15,28,16,29,27,12)(9,18,26,36,25,34,35,22)(14,32,24,30,31,17,33,20)Polycyclic presentation:
2^9|d^2=fh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250469, #72252775, #72252788, … |
| maximal normal subgroup of | #72250469, #72252775, #72252788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411580);