Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(6,7)(8,12)(15,19)(18,20)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(1,2,4,3)(5,6,9,7)(8,10,12,11)(22,26)(23,29)(28,31)(30,35);(2,3)(5,6,9,7)(8,11,12,10)(13,14,17,16)(15,18,19,20)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,23,25,29)(22,27,26,33)(24,30,32,35)(28,34,31,36);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,30)(25,32)(26,31)(27,34)(29,35)(33,36)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=fgi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2501089);