Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42760 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,36)(23,28)(27,30)(29,37)(31,38)(32,39)(33,40)(34,41)(35,43)(42,45)(44,47)(46,48);(1,2)(3,7)(4,8)(5,6)(17,19,23,31)(18,24,28,38)(20,32,35,45)(21,30,33,22)(25,39,43,42)(26,27,40,36)(29,44,46,34)(37,47,48,41);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,33,44,23,35,21,34)(18,25,40,47,28,43,26,41)(19,29,30,45,31,46,22,32)(24,37,27,42,38,48,36,39);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,23,33)(18,26,28,40)(19,30,31,22)(20,34,35,44)(24,27,38,36)(25,41,43,47)(29,45,46,32)(37,42,48,39);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,22,21,31,23,30,33,19)(18,27,26,24,28,36,40,38)(20,32,34,46,35,45,44,29)(25,42,41,37,43,39,47,48)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751023);