Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(7,11)(9,12)(14,17)(18,20)(21,22,26,27)(23,31,34,28)(24,33,35,29)(25,32,36,30);(5,6,8,10)(7,9,11,12)(15,19)(18,20)(22,27)(24,35)(28,31)(30,32);(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,18,19,20)(21,23,26,34)(22,28,27,31)(24,32,35,30)(25,29,36,33);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,24,26,35)(22,29,27,33)(23,32,34,30)(25,28,36,31);(5,7)(6,9)(8,11)(10,12)(21,25,26,36)(22,30,27,32)(23,33,34,29)(24,28,35,31)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514098);