Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437632 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62853 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,23)(16,24)(17,25)(18,27)(19,28)(20,29)(21,30)(22,31)(26,33)(32,35)(34,36);(5,7,10,17)(6,11,14,25)(8,15,19,33)(9,20,22,16)(12,26,28,23)(13,29,31,24)(18,34,21,32)(27,35,30,36);(1,2,3,4)(5,8)(6,12)(7,15)(9,21)(10,19)(11,23)(13,30)(14,28)(16,34)(17,33)(18,22)(20,32)(24,36)(25,26)(27,31)(29,35);(5,8)(6,12)(7,15)(9,18)(10,19)(11,23)(13,27)(14,28)(16,32)(17,33)(20,34)(21,22)(24,35)(25,26)(29,36)(30,31);(5,9,10,22)(6,13,14,31)(7,16,17,20)(8,18,19,21)(11,24,25,29)(12,27,28,30)(15,32,33,34)(23,35,26,36)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349930);