Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
2056 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)(14,17)(15,19)(21,22)(23,29)(24,30)(25,33)(26,27)(28,35)(31,32)(34,36);(5,6,8,10)(7,9,11,12)(22,27)(23,30)(28,33)(31,36);(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,34)(35,36);(1,2,3,4)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,33)(32,35);(5,7,8,11)(6,9,10,12)(21,25,26,35)(22,28,27,33)(23,31,30,36)(24,32,29,34)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342471);