Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1473 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)(19,20)(21,25)(22,26)(23,32)(24,27)(28,30)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,24,29)(18,20,27,31)(21,23,28,22)(25,32,30,26);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,30,24,31,23,25)(18,21,32,29,27,28,26,19);(1,4,6,5)(2,3,7,8)(11,16)(13,15)(17,21,24,28)(18,25,27,30)(19,22,29,23)(20,26,31,32);(1,5,6,4)(2,3,7,8)(10,14)(13,15)(17,22,24,23)(18,26,27,32)(19,28,29,21)(20,30,31,25)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,e^2=fi,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=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6821556);