Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22,24,28)(9,15)(10,14)(11,16)(12,19)(13,23,29,25)(18,32,35,26)(20,27)(21,30)(31,34,36,33);(5,7,9,20,11,21,10,19)(6,12,15,17,16,27,14,30)(8,18,23,34,24,35,25,33)(13,26,22,31,29,32,28,36);(2,4)(5,8,10,25,11,24,9,23)(6,13,15,22,16,29,14,28)(7,18,19,33,21,35,20,34)(12,26,17,31,27,32,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237654, #72237662, #72237669, … |
| maximal normal subgroup of | #72237654, #72237662, #72237669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30657);