Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563319 |
| 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 | |
2850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,13)(7,9)(8,18)(10,11)(14,20)(15,16);(1,2)(3,4)(5,6,10,13,12,16,11,15)(7,17,18,20,8,19,9,14);(5,7,11,9,12,8,10,18)(6,14,13,17,16,20,15,19);(2,4)(5,8,11,18,12,7,10,9)(6,14,13,17,16,20,15,19);(5,9,10,7,12,18,11,8)(6,14,13,17,16,20,15,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213663, #72214877, #72214886, … |
| maximal normal subgroup of | #72213663, #72214877, #72214886, … |
| maximal quotient of | #72226903, #72226907, #72226944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470515);