Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,10)(8,11)(14,17)(15,19)(21,22,26,30)(23,24,28,27)(25,33,35,29)(31,36,34,32);(14,17)(18,20)(21,23,26,28)(22,27,30,24)(25,34,35,31)(29,32,33,36);(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,28)(29,36)(32,33)(34,35);(1,2,4,3)(5,7,9,12)(6,11,10,8)(21,24,26,27)(22,28,30,23)(25,32,35,36)(29,34,33,31);(1,3,4,2)(5,8,9,11)(6,12,10,7)(13,15,16,19)(14,18,17,20)(21,25)(22,29)(23,31)(24,32)(26,35)(27,36)(28,34)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1056216);