Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 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)(3,8)(4,6)(5,7)(17,18)(19,26)(20,23)(21,24)(22,27)(25,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19,21,28)(18,22,24,31)(20,29,30,27)(23,26,25,32);(9,10)(11,12)(13,14)(15,16)(18,23)(20,30)(22,26)(24,25)(27,29)(31,32);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,11,14,16)(18,24)(22,31)(23,25)(26,32)Polycyclic presentation:
2^9|b^2=fg,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6531583);