Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 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)(11,13)(12,14)(15,16)(17,18)(19,27)(20,31)(21,25)(22,36)(23,29)(24,35)(26,33)(28,42)(30,38)(32,46)(34,45)(37,47)(39,44)(40,48)(41,43);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,39)(25,27)(26,38)(28,34)(31,42)(33,43)(35,47)(36,41)(37,40)(44,48)(45,46);(10,14)(11,15)(17,20,33,48,21,34,22,35)(18,24,36,45,25,40,26,31)(19,28,43,47,29,32,30,44)(23,37,41,42,27,39,38,46);(2,5)(6,8)(18,25)(19,29)(20,34)(26,36)(30,43)(35,48)(37,39)(42,46);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(20,34)(24,40)(28,32)(31,45)(35,48)(37,39)(42,46)(44,47)Polycyclic presentation:
2^9|c^2=gh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6671745);