Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501402 |
| 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 | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,28,27,18,9,15,10,14,30,35,11,16,26,32)(7,17,33,36,34,8,20,25,19,13,29,23,21,22,24,31);(1,2)(3,4)(5,7)(6,13)(8,16)(9,24)(10,19)(11,21)(12,29)(14,17)(15,25)(18,22)(20,26)(23,35)(27,34)(28,36)(30,33)(31,32);(2,4)(5,8,11,13,10,22,27,17)(6,7,18,21,14,19,16,34)(9,25,12,23,26,31,30,36)(15,33,35,24,32,29,28,20);(6,14)(7,19)(9,26)(12,30)(16,18)(21,34)(23,36)(25,31)Polycyclic presentation:
2^9|a^2=fh,c^2=eh,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408598);