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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,26)(21,27)(22,37)(23,29)(24,30)(28,38)(31,39)(32,40)(33,41)(34,42)(35,44)(36,45)(43,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,24,32)(18,25,30,40)(20,33,36,47)(21,22,23,38)(26,41,45,43)(27,37,29,28)(31,34,46,35)(39,42,48,44);(2,6)(7,8)(11,16)(13,15)(17,20,23,35,24,36,21,34)(18,26,29,44,30,45,27,42)(19,31,22,33,32,46,38,47)(25,39,37,41,40,48,28,43);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,22,32,38)(20,34,36,35)(25,37,40,28)(26,42,45,44)(31,33,46,47)(39,41,48,43);(1,4,5,3)(2,8,6,7)(17,22,23,32,24,38,21,19)(18,28,29,25,30,37,27,40)(20,33,35,46,36,47,34,31)(26,43,44,39,45,41,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fh,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, 6751291);