Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,12,18)(7,19,23,33,24,32,22,13)(8,25,28,31,21,14,27,34)(9,15,29,35,20,36,26,30);(5,7)(6,13)(8,26)(9,27)(10,22)(11,24)(12,23)(14,15)(16,32)(17,33)(18,19)(20,28)(21,29)(25,36)(30,34)(31,35);(2,4)(5,8)(6,14)(7,20)(9,24)(10,28)(11,21)(12,27)(13,30)(15,19)(16,34)(17,25)(18,31)(22,29)(23,26)(32,36)(33,35);(5,9)(6,15)(7,21)(8,24)(10,29)(11,20)(12,26)(13,31)(14,19)(16,35)(17,36)(18,30)(22,28)(23,27)(25,32)(33,34);(5,10,11,12)(6,16,17,18)(7,22,24,23)(8,27,21,28)(9,29,20,26)(13,32,33,19)(14,31,25,34)(15,35,36,30)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,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 | #72316840 |
| maximal normal subgroup of | #72316840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344998);