Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,11,8,13)(9,14)(10,15)(12,16,19,17)(18,20);(1,2,3,4)(6,11,15,13)(12,14,19,20);(5,7,10,8)(6,12,15,19)(9,17,18,16)(11,14,13,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,f^b=fhi,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72234078, #72234087, … |
| maximal normal subgroup of | #72214565, #72234078, #72234087, … |
| maximal quotient of | #72214449, #72234075, #72238663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53684);