Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2029 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,29)(23,24,31,27)(25,28,35,36)(30,32,33,34);(8,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23,26,31)(22,27,29,24)(25,33,35,30)(28,32,36,34);(6,7)(10,11)(13,14,16,18)(15,17,20,19)(21,24,26,27)(22,23,29,31)(25,34,35,32)(28,33,36,30);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,36)(33,35);(5,8)(6,10)(7,11)(9,12)(13,15,16,20)(14,17,18,19)(21,25,26,35)(22,28,29,36)(23,30,31,33)(24,32,27,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5133262);