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 | C22⋊#95 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,16,10,14)(7,17,20,27,21,30,19,12)(8,22,24,13,25,28,23,29)(18,32,33,31,35,26,34,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,26)(14,30)(15,17)(16,27)(22,31)(23,34)(24,33)(25,35)(28,36)(29,32);(5,8,10,24,11,25,9,23)(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,f^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 | #72226702, #72258198 |
| maximal normal subgroup of | #72226702, #72258198 |
| maximal quotient of | #72226699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30656);