Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,14,16,15)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(10,11)(12,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(19,27)(22,26)(28,32)(30,31);(10,11)(14,15)(18,24)(22,26)(23,25)(30,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6942545);