Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,11)(7,8,12,10)(13,14,16,18)(15,17,19,20)(22,24)(23,29)(25,33)(35,36);(5,7)(6,10)(8,11)(9,12)(15,19)(17,20)(21,22,26,24)(23,30,29,27)(25,32,33,28)(31,35,34,36);(5,7,9,12)(6,8,11,10)(13,15,16,19)(14,17,18,20)(21,23,26,29)(22,27,24,30)(25,34,33,31)(28,36,32,35);(1,2)(3,4)(5,8,9,10)(6,7,11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,22)(23,27,29,30)(25,28,33,32)(31,36,34,35);(5,7)(6,8)(9,12)(10,11)(21,25,26,33)(22,28,24,32)(23,31,29,34)(27,35,30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4042657);