Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1759 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(24,32)(25,27)(26,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18,21,23)(19,27,29,25)(20,24,30,32)(22,31,28,26);(10,14)(13,16)(17,19,21,29)(18,24,23,32)(20,22,30,28)(25,31,27,26);(17,20,21,30)(18,25,23,27)(19,28,29,22)(24,26,32,31);(1,3,4,7)(2,6,5,8)(19,29)(20,30)(24,32)(25,27)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349548);