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 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,17)(15,20,16,19)(21,22,26,29)(23,24,31,27)(25,33,35,28)(30,36,34,32);(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23,26,31)(22,27,29,24)(25,34,35,30)(28,32,33,36);(6,7)(10,11)(13,15,18,16)(14,19,17,20)(21,23)(22,27)(24,29)(25,34)(26,31)(28,32)(30,35)(33,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,17,20)(21,24,26,27)(22,23,29,31)(25,32,35,36)(28,30,33,34);(5,8)(6,10)(7,11)(9,12)(13,17,18,14)(15,19,16,20)(21,25,26,35)(22,28,29,33)(23,30,31,34)(24,32,27,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=g,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, 4944356);