Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1412 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)(17,18,21,27,22,24,23,28)(19,29,37,25,30,42,32,38)(20,33,36,40,35,26,34,41)(31,45,46,48,44,43,47,39);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(18,24)(19,30)(26,33)(27,28)(31,44)(32,37)(40,41)(46,47);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,25,24,38)(20,31,35,44)(21,32,23,37)(26,39,33,48)(27,29,28,42)(34,46,36,47)(40,45,41,43);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,31)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,40)(29,43)(30,44)(32,47)(37,46)(38,48)(42,45);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,28)(19,32,30,37)(20,34,35,36)(25,29,38,42)(26,40,33,41)(31,46,44,47)(39,45,48,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6873842);