Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,12,18)(7,19,21,27)(8,22,25,34)(9,15,13,28)(10,17,11,16)(14,29,33,26)(20,24,36,30)(23,31,32,35);(2,4)(5,7)(6,13)(8,23)(9,18)(10,28)(11,15)(12,21)(14,31)(16,27)(17,19)(20,26)(22,30)(24,34)(25,32)(29,36)(33,35);(5,8,11,26,12,25,10,29)(6,14,17,22,18,33,16,34)(7,20,15,32,21,36,28,23)(9,24,27,31,13,30,19,35);(5,9)(6,15)(7,16)(8,24)(10,19)(11,27)(12,13)(14,32)(17,21)(18,28)(20,34)(22,36)(23,33)(25,30)(26,31)(29,35);(8,25)(14,33)(20,36)(22,34)(23,32)(24,30)(26,29)(31,35)Polycyclic presentation:
2^9|a^2=i,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349912);