Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,19,12,20,9,17,13,16,29,35,21,30,10,18)(7,15,23,14,24,33,8,26,25,32,28,31,27,36,22,34);(1,2,3,4)(5,7,21,24,13,25,12,27)(6,14,30,26,16,31,20,34)(8,10,28,29,22,9,23,11)(15,17,33,19,32,18,36,35);(5,8)(6,15)(7,10)(9,25)(11,27)(12,23)(13,22)(14,17)(16,32)(18,31)(19,26)(20,36)(21,28)(24,29)(30,33)(34,35);(6,16)(14,31)(15,32)(17,18)(19,35)(20,30)(26,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401686);