Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2116 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)(5,6,9,7)(8,10,12,11)(13,14,17,15)(16,19,20,18)(21,22,26,27)(23,31,34,28)(24,29,30,25)(32,33,36,35);(8,12)(10,11)(13,14,17,15)(16,19,20,18)(22,27)(25,29)(28,31)(33,35);(6,7)(10,11)(14,15)(18,19)(21,23,26,34)(22,28,27,31)(24,32,30,36)(25,33,29,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24)(22,29)(23,32)(25,27)(26,30)(28,35)(31,33)(34,36);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,25,26,29)(22,30,27,24)(23,33,34,35)(28,36,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5515791);