Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,23,28,29)(20,21,30,24)(26,27,31,32);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,25,29)(20,27,30,32)(21,31,24,26);(2,6)(7,8)(11,16)(13,15)(17,20,18,24,22,30,25,21)(19,26,29,27,28,31,23,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15);(1,4,5,3)(2,8,6,7)(17,21,25,30,22,24,18,20)(19,27,23,26,28,32,29,31)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751216);