Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2425 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,27)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,23,25,32)(20,28,21,26)(24,27,30,31);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(17,18)(19,27)(20,24)(21,30)(22,25)(23,26)(28,32)(29,31);(17,20,22,21)(18,24,25,30)(19,26,29,28)(23,31,32,27)Polycyclic presentation:
2^9|b^2=fgi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348228);