Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2503 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)(6,7)(8,12)(15,19)(18,20)(21,22)(23,27)(24,31)(25,26)(28,33)(29,36)(30,34)(32,35);(6,7)(10,11)(14,17)(18,20)(22,26)(27,34)(28,31)(35,36);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,30)(22,27,26,34)(24,32,33,29)(28,36,31,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(21,24)(22,28)(23,29)(25,33)(26,31)(27,35)(30,32)(34,36);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,34)(31,33)(32,36)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408264);