Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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×#120 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,20,21,24,31,30)(19,25,28,27,26,23,32,29);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,21,26)(18,23,24,25)(20,29,30,27)(22,28,31,32);(2,6)(5,7)(9,12,13,11)(10,14,15,16)(18,24)(19,26)(20,30)(28,32);(3,8)(5,7)(18,24)(20,30)(23,25)(27,29);(2,6)(5,7)(17,20,22,24,21,30,31,18)(19,27,28,23,26,29,32,25)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312237);