Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,12,7)(8,15)(9,11,14,10)(13,20)(16,19)(17,18);(1,2,4,3)(5,7,10,14,12,6,11,9)(8,16,18,15,20,17,19,13);(5,8)(6,13)(7,15)(9,17)(10,19)(11,18)(12,20)(14,16);(1,3,4,2)(5,9,11,6,12,14,10,7)(8,17,18,13,20,16,19,15)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319366, #72319406, #72319417, … |
| maximal normal subgroup of | #72319366, #72319406, #72319417, … |
| maximal quotient of | #72239934, #72239937, #72239938 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285339);