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)(17,18,22,27,23,28,36,44)(19,29,31,41,32,46,33,24)(20,25,35,43,34,42,48,40)(21,26,38,45,37,39,30,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,32)(18,24,28,41)(20,30,34,38)(21,35,37,48)(22,33,36,31)(25,39,42,26)(27,46,44,29)(40,47,43,45);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,34)(18,25,28,42)(19,30,32,38)(21,33,37,31)(22,35,36,48)(24,39,41,26)(27,43,44,40)(29,47,46,45);(3,8)(5,7)(10,13)(14,16)(17,21,36,30,23,37,22,38)(18,26,44,47,28,39,27,45)(19,20,33,48,32,34,31,35)(24,40,46,42,41,43,29,25);(9,11,12,15)(10,14,13,16)(20,34)(21,37)(25,42)(26,39)(30,38)(35,48)(40,43)(45,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fgi,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=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, 7067889);