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 | |
1456 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)(11,15)(12,14)(13,16)(17,18)(19,25)(20,26)(21,27)(22,38)(23,29)(24,30)(28,39)(31,40)(32,41)(33,42)(34,43)(35,46)(36,44)(37,45)(47,48);(9,11)(10,13)(12,16)(14,15)(17,19,24,32)(18,25,30,41)(20,33,37,35)(21,22,23,39)(26,42,45,46)(27,38,29,28)(31,34,47,36)(40,43,48,44);(2,5)(6,8)(10,14)(11,16)(17,20,23,36,24,37,21,34)(18,26,29,44,30,45,27,43)(19,31,22,33,32,47,39,35)(25,40,38,42,41,48,28,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,22,32,39)(20,34,37,36)(25,38,41,28)(26,43,45,44)(31,33,47,35)(40,42,48,46);(17,22,23,32,24,39,21,19)(18,28,29,25,30,38,27,41)(20,35,36,31,37,33,34,47)(26,42,44,48,45,46,43,40)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=fgi,e^2=fh,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6736043);