Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,15,16,10)(9,12)(13,18)(14,20,19,17);(1,2,3,4)(6,10)(11,17)(12,15)(18,20);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=be,e^a=eh,h^a=hi,f^b=fh,d^c=di,f^e=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72234696, #72234799, #72234834, … |
| maximal normal subgroup of | #72234696, #72234799, #72234834, … |
| maximal quotient of | #72234704, #72236734, #72236736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400083);