Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6,11,15)(7,12,19,28)(8,14,23,31)(9,13,10,16)(17,26,20,27)(18,25,33,35)(21,30,34,36)(22,29,24,32);(1,2,4,3)(5,7)(6,12)(8,21)(9,20)(10,17)(11,19)(13,27)(14,30)(15,28)(16,26)(18,22)(23,34)(24,33)(25,29)(31,36)(32,35);(1,3,4,2)(6,13)(7,17)(9,10)(14,29)(15,16)(18,21)(19,20)(22,24)(26,27)(30,36)(31,32)(33,34);(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,25)(13,29)(15,31)(16,32)(17,21)(19,33)(20,34)(26,30)(27,36)(28,35);(6,15)(7,19)(13,16)(14,31)(17,20)(18,33)(21,34)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346062);