Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,32,41,30,25,33,39)(20,34,37,44,38,26,35,40)(31,42,45,47,36,43,46,48);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,35)(21,23)(25,41)(27,28)(29,39)(31,45)(36,46)(37,38)(40,44)(42,43);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,36)(21,33)(22,30)(23,32)(24,39)(26,43)(27,41)(28,29)(31,38)(34,42)(35,45)(37,46)(40,48)(44,47);(11,16)(13,15)(17,20,23,37,22,38,21,35)(18,26,24,40,28,34,27,44)(19,31,32,45,30,36,33,46)(25,42,39,47,29,43,41,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,21,22,23)(18,27,28,24)(19,32,30,33)(20,37,38,35)(25,39,29,41)(26,40,34,44)(31,46,36,45)(42,48,43,47)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=hi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6995289);