Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 8388608 |
| 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 | |
2230 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)(14,15)(16,20)(21,22,26,27)(23,28,29,24)(25,33,34,30)(31,32,36,35);(5,6,8,10)(7,9,11,12)(13,14,17,15)(16,19,20,18)(22,27)(23,29)(30,33)(31,36);(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,18,20,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36);(1,2)(3,4)(13,16,17,20)(14,18,15,19)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,7)(6,9)(8,11)(10,12)(21,25)(22,30)(23,31)(24,32)(26,34)(27,33)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4360024);