Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,14,10,15,23,11)(7,12,19,28,18,26,16,24)(8,13,21,30,22,29,20,25)(17,27,33,36,32,35,31,34);(6,11)(7,16)(8,20)(9,23)(12,26)(13,29)(14,15)(18,19)(21,22)(27,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,24)(13,27)(14,28)(15,26)(16,23)(20,31)(21,33)(22,32)(25,34)(29,35)(30,36);(5,8)(6,13)(7,17)(9,21)(10,22)(11,25)(12,27)(14,30)(15,29)(16,31)(18,32)(19,33)(20,23)(24,34)(26,35)(28,36);(7,18)(12,26)(16,19)(17,32)(24,28)(27,35)(31,33)(34,36)Polycyclic presentation:
2^9|a^2=f,c^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350230);