Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24271 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
682 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)(7,16)(10,21)(14,31)(17,33)(18,26)(20,24)(22,35)(32,36);(2,4)(5,6,8,13,9,15,11,25)(7,17,26,36,16,33,18,32)(10,22,20,31,21,35,24,14)(12,27,29,19,30,23,28,34);(5,7)(6,12)(8,18)(9,16)(10,23)(11,26)(13,28)(14,32)(15,30)(17,35)(19,20)(21,27)(22,33)(24,34)(25,29)(31,36);(2,4)(5,8,9,11)(6,13,15,25)(7,18,16,26)(10,20,21,24)(12,28,30,29)(14,22,31,35)(17,32,33,36)(19,27,34,23)Polycyclic presentation:
2^9|b^2=fh,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72322902 |
| maximal normal subgroup of | #72322902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409007);