Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| 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)(5,6,9,13,10,15,25,14)(7,16,18,29,20,33,19,32)(8,21,23,12,24,30,36,31)(11,22,26,17,28,34,27,35);(5,7,9,19,10,20,25,18)(6,11,13,27,15,28,14,26)(8,22,23,35,24,34,36,17)(12,29,30,16,31,32,21,33);(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,h^a=hi,d^b=dhi,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240849, #72240858, #72240860, … |
| maximal normal subgroup of | #72240849, #72240858, #72240860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53718);