Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 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)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,30)(20,33)(21,25)(22,27)(23,28)(24,29)(26,37)(31,40)(32,44)(34,42)(35,46)(36,45)(38,41)(39,43)(47,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,34)(21,23)(22,32)(24,38)(26,42)(27,41)(28,30)(29,44)(31,36)(33,48)(35,39)(37,47)(40,43)(45,46);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,39)(25,40)(27,43)(28,33)(29,45)(30,46)(32,47)(34,38)(41,48)(42,44);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(17,21,23,19)(18,25,28,30)(20,35,37,31)(22,32,24,38)(26,40,33,46)(27,44,29,41)(34,36,47,39)(42,45,48,43);(17,22,23,24)(18,27,28,29)(19,32,21,38)(20,36,37,39)(25,41,30,44)(26,43,33,45)(31,47,35,34)(40,48,46,42)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413980);