Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,13,9,15,21,29,10,14,20,30,22,12,27,28)(7,16,18,31,19,24,33,35,17,11,23,25,32,26,34,36);(1,2,3,4)(5,7)(6,11,14,16)(8,18)(9,19)(10,17)(12,24,15,26)(13,25,30,31)(20,23)(21,33)(22,32)(27,34)(28,35,29,36);(6,12,14,15)(8,20)(11,24,16,26)(13,29,30,28)(18,23)(21,27)(25,36,31,35)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,h^2=i,b^a=bd,c^a=ch,f^a=fi,g^a=gi,d^b=di,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224865, #72224874, #72224903, … |
| maximal normal subgroup of | #72224865, #72224874, #72224903, … |
| maximal quotient of | #72224855, #72237204, #72279986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53506);