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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1470 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)(17,18,22,28,23,27,19,24)(20,31,29,39,33,42,36,25)(21,26,34,40,35,41,30,37)(32,44,43,48,46,47,45,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,27,28)(20,29,33,36)(21,34,35,30)(25,31,39,42)(26,40,41,37)(32,45,46,43)(38,47,48,44);(10,13)(14,15)(17,20,23,33)(18,25,27,39)(19,29,22,36)(21,32,35,46)(24,31,28,42)(26,38,41,48)(30,43,34,45)(37,44,40,47);(2,6)(5,7)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,37)(25,38)(27,41)(28,40)(29,43)(31,44)(33,46)(36,45)(39,48)(42,47);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(18,27)(21,35)(24,28)(25,39)(30,34)(31,42)(32,46)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=h,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, 6701152);