Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437452 |
| 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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4630 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)(7,12)(8,13)(9,17)(10,14)(11,15)(16,20)(18,19);(5,7)(6,11)(8,15)(9,18)(10,16)(12,17)(13,20)(14,19);(1,2,3,4)(8,9)(11,19)(13,17)(15,18);(5,7)(6,12)(8,15)(9,18)(10,16)(11,13)(14,20)(17,19);(5,8,10,9)(6,13,14,17)(7,15,16,18)(11,20,19,12)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341765, #72341782 |
| maximal normal subgroup of | #72341765, #72341782 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349765);