Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
7014 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)(5,6,10,15)(7,16,20,25)(8,13,21,28)(9,12,23,14)(11,17,26,19)(18,30,32,36)(22,27,34,29)(24,31,35,33);(2,4)(5,7)(6,11)(8,18)(9,17)(10,20)(12,16)(13,24)(14,25)(15,26)(19,23)(21,32)(22,31)(27,30)(28,35)(29,36)(33,34);(6,12)(7,17)(9,23)(13,27)(14,15)(16,25)(18,31)(19,20)(22,34)(28,29)(30,36)(32,33);(5,8)(6,13)(7,18)(9,22)(10,21)(11,24)(12,27)(14,29)(15,28)(16,30)(17,31)(19,33)(20,32)(23,34)(25,36)(26,35);(8,21)(13,28)(18,32)(22,34)(24,35)(27,29)(30,36)(31,33)Polycyclic presentation:
2^9|a^2=i,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,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338982);