Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2098 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)(5,6,10,11)(7,12,8,9)(13,14,17,20)(15,16,19,18)(21,22,26,30)(23,31,32,27)(24,33,35,28)(25,29,34,36);(6,11)(7,8)(13,15)(14,18)(16,20)(17,19)(21,23,26,32)(22,27,30,31)(24,34,35,25)(28,36,33,29);(5,7,10,8)(6,9,11,12)(13,16,17,18)(14,19,20,15)(21,22)(23,27)(24,28)(25,29)(26,30)(31,32)(33,35)(34,36);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,15)(14,16)(17,19)(18,20)(21,24,26,35)(22,28,30,33)(23,25,32,34)(27,29,31,36);(5,9)(6,8)(7,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,29)(23,24)(26,34)(27,28)(30,36)(31,33)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770302);