Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1356 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,32)(27,36)(29,43)(30,37)(31,41)(33,40)(34,38)(35,42)(39,44)(45,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(24,37)(25,38)(26,27)(28,41)(39,47)(40,42)(43,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,21,32,22,31,36,30)(18,24,27,28,23,37,26,41)(20,29,35,44,34,45,33,46)(25,39,40,43,38,47,42,48);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,34)(18,25,23,38)(19,29,31,45)(21,35,36,33)(24,39,37,47)(26,42,27,40)(28,43,41,48)(30,46,32,44);(10,14)(13,16)(17,21,22,36)(18,26,23,27)(19,30,31,32)(20,33,34,35)(24,28,37,41)(25,40,38,42)(29,44,45,46)(39,48,47,43)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6381276);