Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234437 |
| 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 | #42650 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,40)(32,43)(33,37)(34,41)(35,46)(39,42)(44,48)(45,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,29,34,35)(25,30,40,37)(26,38,41,46)(32,44,45,42)(39,43,48,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,34,45)(24,37,36,30)(26,39,41,48)(29,42,35,44)(38,47,46,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,38)(25,39)(27,41)(28,42)(30,43)(31,44)(33,45)(36,46)(37,47)(40,48);(2,5)(6,8)(10,14)(13,15)(21,34)(26,41)(29,35)(32,45)(38,46)(39,48)(42,44)(43,47)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7143416);