Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,12,17,11)(9,13)(14,19)(15,20,16,18);(1,2,3,4)(5,7)(6,10)(8,16)(9,14)(11,18)(12,20)(13,19)(15,17);(6,11)(8,17)(10,18)(12,13)(15,16)(19,20);(7,14)(10,19)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234146, #72234152, #72234161, … |
| maximal normal subgroup of | #72234146, #72234152, #72234161, … |
| maximal quotient of | #72234139, #72234545, #72261929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401792);