Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2679 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,26,22,21,20,25)(19,27,29,32,28,24,30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(24,27)(29,30)(31,32);(10,13)(14,15)(17,19,23,29,22,28,20,30)(18,24,26,31,21,27,25,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,20,22,23)(18,25,21,26)(19,29,28,30)(24,31,27,32);(1,3,4,7)(2,6,5,8)(17,21,23,25,22,18,20,26)(19,24,29,31,28,27,30,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338825);