Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,40)(20,32,35,44)(21,27,22,26)(25,37,42,34)(29,45,33,48)(30,43,36,39)(38,46,41,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,41)(26,43)(27,39)(28,40)(29,35)(32,48)(34,46)(37,47)(38,42)(44,45);(11,15)(13,16)(17,20,23,35)(18,25,28,42)(19,29,31,33)(21,34,22,37)(24,38,40,41)(26,32,27,44)(30,46,36,47)(39,48,43,45);(2,6)(3,8)(11,15)(13,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,43)(27,39)(28,40)(32,45)(33,35)(34,47)(37,46)(41,42)(44,48);(2,6)(5,7)(10,14)(13,16)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,34,35,37)(24,39,40,43)(25,32,42,44)(29,46,33,47)(38,48,41,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539475);