Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2345 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)(5,6,8,9)(7,10,11,12)(13,14,17,15)(16,19,20,18)(21,22,26,27)(23,29,31,25)(24,32,33,28)(30,34,36,35);(7,11)(10,12)(13,14,17,15)(16,19,20,18)(22,27)(24,33)(25,29)(30,36);(6,9)(10,12)(14,15)(18,19)(21,23,26,31)(22,25,27,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,24,26,33)(22,28,27,32)(23,30,31,36)(25,34,29,35);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,25)(22,23)(24,34)(26,29)(27,31)(28,30)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=fgi,e^2=fgh,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, 4990683);