Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,35)(22,26)(23,27)(25,33)(28,46)(29,40)(31,48)(32,39)(34,45)(36,44)(37,43)(38,42)(41,47);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,24,26,35)(20,31,33,47)(23,36,37,29)(25,41,30,48)(27,44,43,40)(28,38,32,34)(39,45,46,42);(10,14)(13,15)(17,20,22,33)(18,25,26,30)(19,28,21,32)(23,38,37,34)(24,39,35,46)(27,45,43,42)(29,47,36,31)(40,48,44,41);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,21,22,19)(18,24,26,35)(20,32,33,28)(23,36,37,29)(25,39,30,46)(27,40,43,44)(31,34,47,38)(41,45,48,42);(19,21)(24,35)(28,32)(29,36)(31,47)(39,46)(40,44)(41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418354);