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 | #2733 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 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)(9,10)(11,13)(12,14)(15,16)(17,18,24,27,23,29,21,31)(19,22,28,32,30,25,20,26);(1,2)(3,7)(4,8)(5,6)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,21)(22,29)(23,30)(24,28)(26,27)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,23,28)(18,26,29,32)(19,21,30,24)(22,31,25,27);(1,4,5,3)(2,7,6,8)(17,21,23,24)(18,27,29,31)(19,28,30,20)(22,26,25,32);(17,22,24,32,23,25,21,26)(18,28,27,30,29,20,31,19)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292827);