Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,11)(8,19,20,12)(9,14)(10,13)(16,23)(18,28,31,24)(21,32,33,25)(22,26)(27,29)(30,34)(35,36);(2,4)(5,7,9,18,10,17,22,31)(6,11,14,24,13,15,26,28)(8,16,21,30,20,29,33,36)(12,23,25,34,19,27,32,35);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,19)(14,25)(15,27)(17,29)(18,30)(22,33)(24,34)(26,32)(28,35)(31,36);(7,17)(11,15)(16,29)(18,31)(23,27)(24,28)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,g^b=gi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72255602, #72255603, #72255633, … |
| maximal normal subgroup of | #72255602, #72255603, #72255633, … |
| maximal quotient of | #72313308, #72313702, #72313866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260536);