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)(19,31)(20,26)(21,27)(22,28)(23,29)(24,30)(25,34)(32,46)(33,47)(35,45)(36,44)(37,42)(38,43)(39,48)(40,41);(1,3)(2,5)(4,8)(6,7)(17,19)(18,25)(20,35)(21,33)(22,39)(23,34)(24,40)(26,43)(27,41)(28,44)(29,31)(30,47)(32,36)(37,38)(42,45)(46,48);(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,32)(21,36)(22,34)(23,37)(24,39)(25,28)(27,44)(29,42)(30,48)(31,46)(33,38)(35,40)(41,45)(43,47);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,33,34,40)(20,36,37,39)(22,35,32,38)(25,41,31,47)(26,44,42,48)(28,45,46,43);(3,8)(5,7)(17,22,23,32)(18,28,29,46)(19,20,34,37)(21,38,24,35)(25,42,31,26)(27,43,30,45)(33,39,40,36)(41,44,47,48)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6595267);