Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 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)(5,6,9,14,11,16,10,15)(7,17,19,29,21,33,20,36)(8,22,24,31,26,30,25,13)(12,23,27,18,28,34,32,35);(5,7,9,20,11,21,10,19)(6,12,15,27,16,28,14,32)(8,23,24,35,26,34,25,18)(13,29,30,33,31,36,22,17);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239822, #72239835, #72239839, … |
| maximal normal subgroup of | #72239822, #72239835, #72239839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53708);