Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,24,28,25,27,31,34)(23,26,30,33,29,32,35,36);(2,4)(5,7,9,16,10,18,8,17)(6,11,12,19,14,20,13,15)(23,29)(26,32)(30,35)(33,36);(5,6)(7,11)(8,13)(9,12)(10,14)(15,17)(16,19)(18,20)(21,23,24,30,25,29,31,35)(22,26,28,33,27,32,34,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,20,19)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ci,d^b=dh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89686);