Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33580 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,14,10,15,9,19)(7,11,16,20,18,12,17,13);(2,3)(5,7,8,16,10,18,9,17)(6,11,14,20,15,12,19,13);(6,12)(7,16)(8,9)(11,15)(13,19)(14,20)(17,18);(1,3,4,2)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227063, D4×AGL(1,ℤ/16), #72227163, … |
| maximal normal subgroup of | #72227063, D4×AGL(1,ℤ/16), #72227163, … |
| maximal quotient of | #72227052, #72227062, #72227063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405530);