Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(10,11)(14,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,3,5,4)(2,6,7,8)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32);(1,4,5,3)(2,6,7,8)(9,10,12,11)(13,15,16,14)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32);(9,11,12,10)(13,14,16,15)(19,29)(23,26)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6600547);