Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,34,25)(20,26,33,45)(21,38,24,27)(22,28,31,48)(32,41,40,42)(35,47,39,43)(36,46,37,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,34)(18,25,29,30)(20,35,33,39)(21,32,24,40)(22,37,31,36)(26,43,45,47)(27,41,38,42)(28,46,48,44);(10,14)(13,15)(17,20,23,33)(18,26,29,45)(19,31,34,22)(21,37,24,36)(25,28,30,48)(27,46,38,44)(32,39,40,35)(41,43,42,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,38)(19,32,34,40)(20,36,33,37)(22,39,31,35)(25,41,30,42)(26,44,45,46)(28,47,48,43);(2,5)(6,8)(11,16)(13,15)(17,22,23,31)(18,28,29,48)(19,33,34,20)(21,39,24,35)(25,26,30,45)(27,47,38,43)(32,37,40,36)(41,44,42,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606333);