Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1470 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,23)(19,25,27,22)(21,24,29,31)(26,30,28,32);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(10,13)(14,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(18,23)(19,27)(24,31)(26,28);(1,5,6,4)(2,3,7,8)(19,27)(22,25)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7420712);