Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| 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×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3549 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,13,11,16,9,15)(7,14,18,20,8,12,17,19);(6,12)(7,8)(9,10)(13,19)(14,16)(15,20);(2,4)(6,13)(9,10)(12,19)(14,20)(15,16)(17,18);(5,7,11,8)(6,14,16,12)(9,17,10,18)(13,20,15,19)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238617, #72261005, #72261017, … |
| maximal normal subgroup of | #72238617, #72261005, #72261017, … |
| maximal quotient of | #72238625, #72250630, #72261960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400810);