Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1730 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)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,23,13,25,29,24,30)(18,26,33,35,34,36,32,31);(2,4)(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,23,33,25,34,24,32)(13,26,29,35,30,36,22,31);(5,8)(6,13,16,30)(7,18)(9,24)(10,23)(11,25)(12,26,28,36)(14,29,15,22)(17,31,27,35)(19,33)(20,32)(21,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,d^b=di,e^b=ei,d^c=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250586, #72250596, #72250605, … |
| maximal normal subgroup of | #72250586, #72250596, #72250605, … |
| maximal quotient of | C2×#123505 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30701);