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 | |
1512 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)(17,18,23,27,22,26,21,25)(19,28,29,32,30,20,24,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,23,29,22,30,21,24)(18,20,27,31,26,28,25,32);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,28)(18,24,26,29)(19,25,30,27)(21,32,23,31);(2,6)(7,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,32)(22,28)(23,31)(25,30)(26,29);(1,4,5,3)(2,8,6,7)(17,21,22,23)(18,25,26,27)(19,29,30,24)(20,31,28,32)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,c^b=ci,d^b=dfi,e^b=ei,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, 6290900);