Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 | |
5085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,12)(7,16)(9,10)(13,18)(14,15)(17,20);(1,2,3,4)(5,6,9,12,11,14,10,15)(7,17,16,19,18,20,13,8);(5,7,10,16,11,18,9,13)(6,8,15,17,14,19,12,20);(5,8)(6,13)(7,15)(9,20)(10,17)(11,19)(12,18)(14,16);(6,14)(7,18)(12,15)(13,16)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213663, #72213751, … |
| maximal normal subgroup of | #72213527, #72213663, #72213751, … |
| maximal quotient of | #72224689, #72226238, #72226928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350492);