Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1455 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)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,31)(21,35)(22,28)(23,24)(25,32)(26,37)(29,41)(30,39)(33,46)(34,38)(36,42)(40,43)(44,47)(45,48);(1,3,7,4)(2,5,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,37,36)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(1,4,7,3)(2,5,8,6)(10,13)(14,15)(17,20,19,29,22,34,23,32)(18,25,24,38,28,41,27,31)(21,33,30,43,37,47,36,45)(26,40,39,46,35,48,42,44);(1,5,7,6)(2,3,8,4)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,36)(24,39)(25,40)(27,42)(28,35)(29,43)(31,44)(32,45)(34,47)(38,46)(41,48);(1,6,7,5)(2,3,8,4)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,36,37,30)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,46,48,44)Polycyclic presentation:
2^9|a^2=hi,b^2=fhi,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893122);