Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,14)(8,19)(9,12)(10,13)(11,17)(15,18)(16,20);(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,20)(14,19);(1,2,3,4)(5,6)(7,15)(8,12)(9,19)(10,13)(11,17)(14,18)(16,20);(5,6)(7,15)(8,12)(9,19)(10,13)(11,17)(14,18)(16,20);(5,8,10,9)(6,12,13,19)(7,16,18,17)(11,15,20,14)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215857, #72215974, #72239257, … |
| maximal normal subgroup of | #72215857, #72215974, #72239257, … |
| maximal quotient of | #72239266, C2×#6442607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349803);