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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2406 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)(5,6,9,12)(7,11,8,10)(15,19)(18,20)(22,24)(27,31)(28,32)(34,35);(7,8)(10,11)(14,17)(18,20)(21,22,26,24)(23,27,30,31)(25,32,33,28)(29,35,36,34);(5,7,9,8)(6,10,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(21,24,26,22)(23,27,30,31)(25,32,33,28)(29,34,36,35);(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,33)(22,28,24,32)(23,29,30,36)(27,34,31,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=fg,e^2=f,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, 4374232);