Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,8,7)(9,10,13,12)(11,15,16,14)(20,27)(24,29)(25,31)(30,32)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(2,5)(6,7)(9,11,13,16)(10,14,12,15)(18,22)(23,28)(24,29)(30,32)(34,39)(35,41)(42,45)(43,48);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,45)(46,48);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4370621);