Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,27,30,25)(20,22,21,24)(28,29,32,31);(1,3,5,8)(2,6,7,4)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,25,26,27)(20,28,21,32)(22,31,24,29);(10,14)(13,16)(17,20,23,21)(18,22,26,24)(19,27,30,25)(28,31,32,29);(1,4,5,6)(2,3,7,8)(10,14)(13,16)(17,21,23,20)(18,22,26,24)(19,28,30,32)(25,29,27,31);(17,22,23,24)(18,20,26,21)(19,29,30,31)(25,32,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412238);