Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1544 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)(11,15,16,13)(17,18)(19,30)(20,34)(21,38)(22,28)(23,29)(24,27)(25,33)(26,37)(31,42)(32,43)(35,45)(36,46)(39,48)(40,47)(41,44);(1,3)(2,5)(4,8)(6,7)(17,19)(18,25)(20,35)(21,32)(22,40)(23,33)(24,41)(26,45)(27,43)(28,46)(29,30)(31,36)(34,48)(37,39)(38,44)(42,47);(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,36)(22,33)(23,37)(24,40)(25,42)(27,46)(28,30)(29,34)(32,39)(35,41)(38,47)(43,45)(44,48);(2,6)(5,7)(17,21,23,24)(18,27,29,38)(19,32,33,41)(20,36,37,40)(22,35,31,39)(25,43,30,44)(26,46,34,47)(28,48,42,45);(3,8)(5,7)(17,22,23,31)(18,28,29,42)(19,20,33,37)(21,39,24,35)(25,26,30,34)(27,45,38,48)(32,40,41,36)(43,47,44,46)Polycyclic presentation:
2^9|a^2=gi,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595111);