Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #42778 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1441 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)(11,15)(13,16)(17,18,21,26,22,27,34,24)(19,23,29,37,30,38,45,35)(20,31,33,39,32,25,40,41)(28,42,44,46,43,36,47,48);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19,22,30)(18,23,27,38)(20,28,32,43)(21,29,34,45)(24,35,26,37)(25,36,31,42)(33,44,40,47)(39,46,41,48);(9,11,12,15)(10,13,14,16)(18,24)(21,34)(23,35)(25,39)(26,27)(29,45)(31,41)(33,40)(36,46)(37,38)(42,48)(44,47);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,32)(18,25,27,31)(19,28,30,43)(21,33,34,40)(23,36,38,42)(24,39,26,41)(29,44,45,47)(35,46,37,48);(2,5)(6,8)(10,14)(13,16)(20,32)(25,31)(28,43)(33,40)(36,42)(39,41)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7180570);