Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1430 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)(17,18,21,24)(19,25,27,23)(20,22,30,31)(26,28,29,32);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(17,19,21,27)(18,23,24,25)(20,29,30,26)(22,28,31,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,24,21,30,31,18)(19,26,28,23,27,29,32,25);(1,4,6,5)(2,3,7,8)(10,13)(14,16)(18,24)(19,27)(20,30)(28,32);(1,5,6,4)(2,3,7,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=f,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=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6878411);