Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,10)(7,11,12,8)(15,19)(18,20)(21,22,26,28)(23,29,31,24)(25,27,34,36)(30,35,33,32);(6,10)(8,11)(14,17)(18,20)(21,23,26,31)(22,24,28,29)(25,33,34,30)(27,35,36,32);(5,7,9,12)(6,11,10,8)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,30)(26,31)(27,32)(28,29)(33,34)(35,36);(1,2)(3,4)(13,15,16,19)(14,18,17,20)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,36)(34,35);(5,8,9,11)(6,12,10,7)(21,25,26,34)(22,27,28,36)(23,30,31,33)(24,32,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1943946);