Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,23,27)(19,24,29,38)(20,32,35,41)(21,26,22,36)(25,33,40,34)(28,42,43,48)(30,39,31,46)(37,44,47,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,45)(35,43)(36,46)(40,47)(41,48);(3,8)(5,7)(10,13)(14,16)(17,20,23,35)(18,25,27,40)(19,28,29,43)(21,34,22,33)(24,37,38,47)(26,41,36,32)(30,45,31,44)(39,48,46,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,46)(42,48)(44,45);(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,46)(25,32,40,41)(28,44,43,45)(37,42,47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678613);