Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1882 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)(5,6)(7,11)(8,12)(9,10)(14,18)(16,19)(24,31)(28,33)(29,35)(34,36);(5,7,9,12)(6,8,10,11)(14,18)(15,20)(22,26)(27,32)(28,33)(34,36);(5,6,9,10)(7,11,12,8)(13,14,17,18)(15,19,20,16)(21,22,25,26)(23,27,30,32)(24,28,31,33)(29,34,35,36);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,6)(7,8)(9,10)(11,12)(13,16,17,19)(14,15,18,20)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=gh,e^2=fg,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, 4517199);