Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1861 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)(7,10)(8,11)(9,12)(14,17)(15,19)(21,22,26,29)(23,25,31,27)(24,32,34,28)(30,33,36,35);(5,7)(6,8)(9,11)(10,12)(21,23,26,31)(22,27,29,25)(24,30,34,36)(28,33,32,35);(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,30)(26,31)(27,29)(28,35)(32,33)(34,36);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,34)(22,28,29,32)(23,30,31,36)(25,35,27,33);(5,8,9,10)(6,11,12,7)(21,25)(22,23)(24,33)(26,27)(28,36)(29,31)(30,32)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gi,e^2=fh,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, 4342538);