Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(10,13)(11,15)(17,18,22,19,21,24,31,26)(20,27,30,28,29,23,32,25);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,24,21,26,31,18)(20,28,30,23,29,25,32,27);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32);(2,6)(7,8)(10,13)(14,16)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32);(1,4,5,3)(2,8,6,7)(20,29)(23,27)(25,28)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290892);