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) = C26 |
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 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(26,29)(30,31);(1,3,5,8)(2,6,7,4)(10,12)(14,15)(18,22)(19,24)(25,26,28,29)(27,31,32,30);(2,7)(3,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24);(1,4,5,6)(2,8,7,3)(9,12,13,10)(11,15,16,14)(17,18)(19,20)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,e^2=fh,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, 4862841);