Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1439 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)(17,18,22,27)(19,28,30,32)(20,25,21,26)(23,29,24,31);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,29,22,30,20,31)(18,23,25,32,27,24,26,28);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,20,31,22,30,21,29)(18,24,26,32,27,23,25,28);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,25,27,26)(19,29,30,31)(23,28,24,32);(1,4,5,3)(2,6,8,7)(19,30)(23,24)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372580);