Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,21,27,22,26,34,24)(19,23,29,38,30,37,44,35)(20,25,33,41,32,42,46,39)(28,36,43,48,31,40,45,47);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,37)(20,31,32,28)(21,29,34,44)(24,35,27,38)(25,40,42,36)(33,45,46,43)(39,48,41,47);(9,11,12,16)(10,13,14,15)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(1,3,5,4)(2,6,7,8)(17,20,22,32)(18,25,26,42)(19,28,30,31)(21,33,34,46)(23,36,37,40)(24,39,27,41)(29,43,44,45)(35,47,38,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(20,32)(23,37)(24,27)(28,31)(33,46)(35,38)(43,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6668266);