Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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 | |
4504 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)(6,12)(7,16)(9,10)(13,28)(14,27)(15,23)(18,31)(19,24)(20,30)(21,25)(29,36)(33,34);(2,4)(5,6)(7,17)(8,20)(9,23)(10,12)(11,15)(13,16)(14,21)(18,29)(19,26)(22,27)(24,28)(25,30)(31,35)(32,33)(34,36);(5,7,10,16,11,19,9,24)(6,13,12,26,15,28,23,17)(8,18,21,31,22,34,25,33)(14,29,27,35,30,36,20,32);(5,8)(6,14)(7,18)(9,25)(10,21)(11,22)(12,27)(13,29)(15,30)(16,31)(17,32)(19,34)(20,23)(24,33)(26,35)(28,36);(6,15)(12,23)(13,28)(14,30)(17,26)(20,27)(29,36)(32,35)Polycyclic presentation:
2^9|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,e^b=ei,f^b=fi,h^b=hi,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, 6349906);