Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2144 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)(7,11)(9,12)(15,19)(16,18)(21,22)(23,24)(25,33)(26,27)(28,29)(30,35)(31,34)(32,36);(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,16,19,18)(22,27)(23,29)(30,33)(31,36);(5,7)(6,9)(8,11)(10,12)(13,15,17,19)(14,18,20,16)(21,23)(22,28)(24,27)(25,31)(26,29)(30,34)(32,33)(35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,28)(22,29,27,23)(25,34,35,32)(30,31,33,36);(5,7)(6,9)(8,11)(10,12)(13,16,17,18)(14,19,20,15)(21,25,26,35)(22,30,27,33)(23,31,29,36)(24,32,28,34)Polycyclic presentation:
2^9|b^2=fg,c^2=fh,d^2=f,e^2=h,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, 2645636);