Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| 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 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,16)(8,13)(9,25)(10,14)(11,15)(12,19)(17,33)(18,21)(20,28)(22,27)(23,30)(24,32)(26,31)(29,34)(35,36);(5,7,11,21)(6,12,15,28)(8,22,23,35)(9,20,10,19)(13,29,30,33)(14,18,25,16)(17,26,34,24)(27,31,36,32);(1,2,3,4)(5,8)(6,13)(7,17)(9,24)(10,26)(11,23)(12,27)(14,31)(15,30)(16,33)(18,29)(19,22)(20,35)(21,34)(25,32)(28,36);(5,6)(7,18)(8,13)(9,14)(10,25)(11,15)(12,19)(16,21)(17,29)(20,28)(22,27)(23,30)(24,31)(26,32)(33,34)(35,36);(8,23)(13,30)(17,34)(22,35)(24,26)(27,36)(29,33)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318643, #72319042, #72341707, … |
| maximal normal subgroup of | #72318643, #72319042, #72341707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349808);