Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2550 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)(7,16)(8,20,22,13)(9,23)(10,14)(11,15)(12,18)(17,33,35,32)(19,28)(21,30,25,31)(24,27)(26,36,29,34);(5,7,10,18,11,19,9,24)(6,12,14,28,15,27,23,16)(8,17,21,36,22,35,25,34)(13,29,31,32,20,26,30,33);(2,4)(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,31)(15,20)(16,32)(18,36)(19,35)(23,30)(24,34)(27,29)(28,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237702, #72237704, #72237711, … |
| maximal normal subgroup of | #72237702, #72237704, #72237711, … |
| maximal quotient of | #72281067, #72281324, #72303292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30782);