Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2431 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,28,23,19,22,27)(20,29,21,26,31,25,32,30);(1,3,6,2)(4,7,5,8)(17,19,24,27,23,18,22,28)(20,25,21,30,31,29,32,26);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,24)(22,32)(23,31)(26,27)(28,30);(1,5,6,4)(2,7,3,8)(17,21)(18,26)(19,30)(20,22)(23,32)(24,31)(25,28)(27,29);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,19,28)(20,21,31,32)(25,30,29,26)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^c=di,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, 6284670);