Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501060 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12,19,23,34,33,8,14,18,28,10,13,22,21)(7,11,16,25,29,32,36,35,15,24,20,27,17,26,31,30);(1,2)(3,4)(5,7)(6,11)(8,15)(9,20)(10,17)(12,27)(13,26)(14,24)(16,18)(19,29)(21,35)(22,36)(23,32)(25,28)(30,33)(31,34);(2,4)(5,6)(7,11)(8,14)(9,21)(10,23)(12,22)(13,19)(15,24)(16,30)(17,32)(18,33)(20,35)(25,31)(26,29)(27,36)(28,34);(7,15)(9,18)(11,24)(12,28)(17,29)(21,33)(22,34)(26,32)Polycyclic presentation:
2^9|a^2=f,f^2=hi,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, 408256);