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 | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,19,35)(8,20,23,33)(9,24,25,36)(10,27,21,15)(12,28,29,18)(13,31,30,22)(14,32,34,26);(1,2)(3,4)(5,7)(6,12)(8,21)(9,22)(10,23)(11,19)(13,15)(14,33)(16,29)(17,24)(18,26)(20,34)(25,31)(27,30)(28,32)(35,36);(2,4)(9,25)(14,34)(17,35)(18,28)(20,33)(22,31)(24,36)(26,32);(5,8,10,7,11,23,21,19)(6,13,15,29,16,30,27,12)(9,18,26,31,25,28,32,22)(14,35,24,33,34,17,36,20)Polycyclic presentation:
2^9|a^2=i,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 | #72321457 |
| maximal normal subgroup of | #72321457 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411582);