Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #24455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,34)(28,45)(29,39)(30,40)(32,48)(33,44)(35,41)(36,43)(38,47)(42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,39)(20,32,33,28)(21,35,22,30)(25,42,43,38)(26,41,37,40)(31,48,44,45)(34,46,36,47);(2,6)(5,7)(10,13)(14,16)(17,20,22,34,23,33,21,36)(18,25,26,31,27,43,37,44)(19,28,30,47,29,32,35,46)(24,38,41,45,39,42,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(20,33)(24,39)(25,43)(30,35)(31,44)(34,36)(40,41);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,35)(20,34,33,36)(24,40,39,41)(25,44,43,31)(28,46,32,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6692043);