Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
2062 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)(5,6,9,10)(7,11,12,8)(13,14,17,20)(15,16,19,18)(21,22)(23,30)(24,28)(25,27)(26,33)(29,34)(31,35)(32,36);(6,10)(8,11)(13,14)(15,16)(17,20)(18,19)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,16)(17,19)(18,20)(21,25,27,22)(23,30,24,28)(26,29,34,33)(31,35,32,36);(5,8,9,11)(6,12,10,7)(13,16)(14,19)(15,20)(17,18)(21,26,27,34)(22,29,25,33)(23,31,24,32)(28,35,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5529825);