Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,29)(30,42)(32,41)(33,47)(34,39)(35,38)(36,48)(37,44)(40,45)(43,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,37,30)(25,39,41,31)(26,42,44,38)(33,45,36,46)(40,47,43,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,36,37,33)(24,31,28,39)(26,43,44,40)(30,46,35,45)(38,48,42,47);(3,8)(5,7)(11,16)(13,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,35)(24,38)(25,40)(27,44)(28,42)(29,45)(31,47)(32,46)(34,36)(39,48)(41,43);(10,14)(13,15)(18,27)(21,37)(24,28)(25,41)(30,35)(31,39)(33,36)(45,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901811);