Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1803 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)(5,6,9,11)(7,10,12,8)(13,14,18,16)(15,20,17,19)(21,22)(23,28)(24,32)(25,34)(26,27)(29,30)(31,35)(33,36);(5,7,9,12)(6,10,11,8)(22,27)(25,32)(28,35)(30,36);(6,11)(7,12)(13,15,18,17)(14,19,16,20)(21,23)(22,28)(24,33)(25,30)(26,31)(27,35)(29,34)(32,36);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16,18,14)(15,19,17,20)(21,24)(22,25)(23,29)(26,34)(27,32)(28,36)(30,35)(31,33);(5,8,9,10)(6,12,11,7)(13,17,18,15)(14,19,16,20)(21,25)(22,24)(23,30)(26,32)(27,34)(28,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843857);