Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(1,3,4,8)(2,5,6,7)(17,19,21,29)(18,23,24,27)(20,30,22,28)(25,32,26,31);(2,6)(5,7)(10,13)(14,16)(19,28)(20,22)(23,31)(25,26)(27,32)(29,30);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,24)(23,27)(25,26)(31,32);(3,8)(5,7)(17,20)(18,25)(19,29)(21,22)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6607666);