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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1394 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)(11,15)(14,16)(17,18,22,27,23,28,38,46)(19,29,31,41,32,45,35,24)(20,25,34,43,33,42,48,40)(21,26,37,44,36,39,30,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,32)(18,24,28,41)(20,30,33,37)(21,34,36,48)(22,35,38,31)(25,39,42,26)(27,45,46,29)(40,47,43,44);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,33)(18,25,28,42)(19,30,32,37)(21,35,36,31)(22,34,38,48)(24,39,41,26)(27,43,46,40)(29,47,45,44);(3,8)(5,7)(10,13)(14,16)(17,21,22,37,23,36,38,30)(18,26,27,44,28,39,46,47)(19,20,31,34,32,33,35,48)(24,40,29,25,41,43,45,42);(9,11,12,15)(10,14,13,16)(20,33)(21,36)(25,42)(26,39)(30,37)(34,48)(40,43)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7067919);