Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2204 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,32)(21,27)(22,24)(23,36)(25,30)(26,34)(29,44)(31,40)(33,42)(35,41)(37,38)(39,45)(43,46)(47,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,37)(24,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,46)(35,47)(41,43)(42,44);(2,6)(5,7)(17,20,22,34)(18,26,24,32)(19,29,31,46)(21,35,23,33)(25,39,38,48)(27,42,36,41)(28,43,40,44)(30,47,37,45);(3,8)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(29,45,46,47)(39,43,48,44)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6652939);