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 | |
2478 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)(5,6,9,15,11,14,10,24)(7,12,19,28,17,27,18,26)(8,20,22,30,21,31,23,13)(16,32,34,25,35,36,33,29);(2,4)(5,7,10,19,11,17,9,18)(6,12,15,26,14,27,24,28)(8,16,22,33,21,35,23,34)(13,29,31,32,30,25,20,36);(5,8)(6,13)(7,16)(9,23)(10,22)(11,21)(12,25)(14,30)(15,31)(17,35)(18,34)(19,33)(20,24)(26,36)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^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 | #72237174, #72237178, #72237189, … |
| maximal normal subgroup of | #72237174, #72237178, #72237189, … |
| maximal quotient of | #72239032, #72239709, C2×#123611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30807);