Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,10,9,11,7,12)(13,14,16,19,17,15,18,20);(1,2,3,4)(6,10)(7,8)(11,12)(13,15,16,20,17,14,18,19);(6,10)(7,8)(11,12);(6,11)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#3003, #72310284, #72310286, … |
| maximal normal subgroup of | C8×#3003, #72310284, #72310286, … |
| maximal quotient of | #72286299, C2×#193537 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100733);