Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533087 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(18,22)(23,25)(24,29)(30,31);(1,3,5,4)(2,6,8,7)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3,5,4)(2,7,8,6)(10,13)(11,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,3)(2,6,8,7)(20,28)(24,29)(26,32)(30,31);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^c=gi,e^d=eh89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6448375);