Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1867 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,19)(3,13,16,23,15,29,20,30)(4,10,17,9,18,26,21,27)(8,22,25,31,24,32,28,14);(1,3,6,16,7,15,5,20)(2,8,11,25,12,24,19,28)(4,14,17,22,18,31,21,32)(9,23,26,29,27,30,10,13);(1,4)(2,9)(3,14)(5,21)(6,17)(7,18)(8,23)(10,19)(11,26)(12,27)(13,28)(15,31)(16,22)(20,32)(24,30)(25,29);(3,15)(8,24)(13,29)(14,31)(16,20)(22,32)(23,30)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236884, #72237551, #72237553, … |
| maximal normal subgroup of | #72236884, #72237551, #72237553, … |
| maximal quotient of | #72312075, #72337477, C2×#494379 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401575);