Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,11,15,10,16,9,14)(7,17,19,13,18,20,8,12);(1,2,4,3)(5,7,9,8,10,18,11,19)(6,12,15,17,16,13,14,20);(1,2,4,3)(5,8,9,18,10,19,11,7)(6,13,15,20,16,12,14,17);(1,3,4,2)(6,14,16,15)(12,20,13,17)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216206, #72249991, #72257156, … |
| maximal normal subgroup of | #72216206, #72249991, #72257156, … |
| maximal quotient of | #72215689, #72215706, #72216075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283459);