Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6,8,11,10,13,9,14)(7,12,16,15,18,19,17,20);(2,4)(5,6)(7,15)(8,14)(9,11)(10,13)(12,16)(17,19)(18,20);(6,11)(8,9)(12,15)(13,14)(16,17)(19,20);(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,19)(14,20);(6,13)(11,14)(12,19)(15,20)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213751, #72213818, … |
| maximal normal subgroup of | #72213527, #72213751, #72213818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345111);