Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34880 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 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)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,22,26,21,25)(19,20,29,31,28,30,24,32);(1,3,6,2)(4,7,5,8)(17,19,23,29,22,28,21,24)(18,20,27,31,26,30,25,32);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,32)(22,30)(23,31)(25,28)(26,29);(1,5,6,4)(2,7,3,8)(19,28)(20,30)(24,29)(31,32);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,29,28,24)(20,31,30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284656);