Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | #30396 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1494 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,27,30,24)(20,31,28,25)(22,26,29,32);(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,30)(22,29)(24,25)(27,31);(2,6)(7,8)(10,14)(11,16)(17,19,29,20,21,30,22,28)(18,24,32,25,23,27,26,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(19,30)(20,28)(26,32);(1,4,5,3)(2,8,6,7)(17,20,29,30,21,28,22,19)(18,25,32,27,23,31,26,24)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749345);