Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,30,32,24)(20,34,33,25)(21,29,35,26)(22,38,40,27)(31,41,47,46)(36,42,39,44)(37,43,48,45);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,35,33,21)(22,36,40,39)(25,26,34,29)(27,42,38,44)(31,37,47,48)(41,45,46,43);(2,6)(7,8)(10,14)(11,16)(17,20,21,19,23,33,35,32)(18,25,29,24,28,34,26,30)(22,39,37,47,40,36,48,31)(27,44,45,46,38,42,43,41);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,40,48)(24,34,30,25)(27,43,38,45)(31,36,47,39)(41,44,46,42);(1,4,5,3)(2,8,6,7)(17,22,35,48,23,40,21,37)(18,27,26,43,28,38,29,45)(19,31,20,36,32,47,33,39)(24,41,25,42,30,46,34,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=fgi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749327);