Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #385 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,7)(8,12,10,11)(13,14,15,17,16,18,19,20);(5,7,8,12,9,6,10,11);(1,2,3,4)(5,6)(7,9)(8,11)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,f^c=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214091, #72214561, #72215902, … |
| maximal normal subgroup of | #72214091, #72215902, #72215904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28613);