Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#447 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14,16,15,17,18,20,19);(6,10)(7,8)(11,12);(2,4);(5,7,9,8)(6,11,12,10)(13,15,16,18,17,19,20,14)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72286025, #72318326, #72318337, … |
| maximal normal subgroup of | #72286025, #72318326, #72318337, … |
| maximal quotient of | #72286064, #72286168, #72286292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 115870);