Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,12,19,26,18,27,24,31)(8,20,23,13,21,30,22,29)(17,32,33,28,34,36,35,25);(1,2)(3,4)(5,7)(6,12,14,27)(8,17)(9,19)(10,24)(11,18)(13,28,29,25)(15,26,16,31)(20,36,30,32)(21,34)(22,33)(23,35);(2,4)(5,8)(6,13)(7,17)(9,23)(10,22)(11,21)(12,25)(14,29)(15,30)(16,20)(18,34)(19,35)(24,33)(26,36)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237696, #72237703, #72237713, … |
| maximal normal subgroup of | #72237696, #72237703, #72237713, … |
| maximal quotient of | #72226734, #72280450, #72280875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30806);