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 | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1565 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)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,21,29,24)(20,25,28,32)(22,30,31,26);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,25)(21,27)(22,30)(23,29)(26,31)(28,32);(10,14)(13,15)(17,20,23,28)(18,25,27,32)(19,22,29,31)(21,30,24,26);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,23,24)(18,19,27,29)(20,26,28,30)(22,25,31,32);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(17,22)(18,26)(19,28)(20,29)(21,25)(23,31)(24,32)(27,30)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6649853);