Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1601 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18,21,23)(19,26,24,20)(22,25,29,30)(27,28,32,31);(1,3)(2,5)(4,8)(6,7)(17,18)(19,27)(20,28)(21,23)(22,30)(24,32)(25,29)(26,31);(10,13)(14,16)(17,19,21,24)(18,20,23,26)(22,31,29,28)(25,32,30,27);(2,6)(5,7)(18,23)(20,26)(25,30)(27,32);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,23)(21,26)(22,32)(25,28)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6620760);