Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2028 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)(6,8)(7,11)(13,14,16,18)(15,17,19,20)(21,22,26,29)(23,30,27,24)(25,33,35,28)(31,32,36,34);(7,11)(10,12)(15,19)(17,20)(21,23,26,27)(22,24,29,30)(25,31,35,36)(28,32,33,34);(5,6,9,8)(7,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,24)(25,31)(26,27)(28,32)(29,30)(33,34)(35,36);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,24)(22,27)(23,29)(25,34)(26,30)(28,31)(32,35)(33,36);(5,8,9,6)(7,12,11,10)(21,25,26,35)(22,28,29,33)(23,31,27,36)(24,32,30,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2643639);