Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | #39386 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(17,19,21,18)(20,23,28,25)(22,26,30,24)(27,31,32,29);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(1,4,6,5)(2,3,7,8)(10,13)(14,16)(18,19)(23,25)(24,26)(27,32);(1,5,6,4)(2,3,7,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6878382);