Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,16)(18,19)(21,22,26,25)(23,29,31,27)(24,28,32,33)(30,35,36,34);(8,12)(10,11)(13,14,17,16)(15,19,20,18)(22,25)(24,32)(27,29)(30,36);(6,7)(10,11)(14,16)(15,20)(21,23,26,31)(22,27,25,29)(24,30,32,36)(28,34,33,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,18,16,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,19,20,18)(21,25,26,22)(23,27,31,29)(24,33,32,28)(30,34,36,35)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4950047);