Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,12,16,14,18,13,17)(11,19,22,24,20,23,21,15);(1,2,4,7,6,8,5,3)(9,11,13,21,14,20,12,22)(10,15,17,23,18,24,16,19);(1,3,4,2,6,7,5,8)(10,16,18,17)(11,20)(15,23,24,19)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72342987, #72342988, #72343008, … |
| maximal normal subgroup of | #72342987, #72342988, #72343008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17653);