Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6)(7,15)(8,18)(9,19)(10,14)(11,13)(12,17)(16,20);(1,2,4,3)(5,7,10,17)(6,12,14,15)(8,19,20,13)(9,18,11,16);(1,3,4,2)(5,8,10,20)(6,7,14,17)(9,12,11,15)(13,16,19,18);(5,6)(7,8)(9,13)(10,14)(11,19)(12,16)(15,18)(17,20);(5,9,10,11)(6,13,14,19)(7,16,17,18)(8,12,20,15)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213527, #72213818, … |
| maximal normal subgroup of | #72213480, #72213527, #72213818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346063);