Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,14)(9,25)(10,27)(11,16)(12,20)(13,19)(15,23)(18,33)(21,28)(22,34)(24,32)(26,29)(30,35)(31,36);(1,2,4,3)(7,18)(8,19)(9,24)(10,23)(12,28)(13,14)(15,27)(17,33)(20,21)(22,26)(25,32)(29,34)(30,31)(35,36);(5,7,19,18,10,21,24,35,11,22,9,26,23,36,8,20)(6,12,14,31,15,29,25,34,16,30,32,28,27,33,13,17);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410347);