Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,17,12,18,11,16)(7,8,9,15,14,20,19,13);(1,2,4,3)(5,7,12,14)(6,13,18,15)(8,16,20,17)(9,11,19,10);(1,3,4,2)(5,8,10,15,12,20,11,13)(6,14,17,19,18,7,16,9);(5,9,11,7,12,19,10,14)(6,15,16,8,18,13,17,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,e^b=ei,d^c=di,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72213302, #72213416, … |
| maximal normal subgroup of | #72213250, #72213302, #72213416, … |
| maximal quotient of | #72210005, #72239936, #72239938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264325);