Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,24)(21,25)(22,27)(23,31)(26,29)(30,42)(32,37)(33,39)(34,41)(35,40)(36,45)(38,43)(44,47)(46,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,29,43,22,32,21,31)(18,23,25,37,27,38,26,28)(20,30,33,44,35,46,34,45)(24,36,41,48,40,47,39,42);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20,22,35)(18,24,27,40)(19,30,32,46)(21,34,29,33)(23,36,38,47)(25,41,26,39)(28,42,37,48)(31,45,43,44);(10,14)(13,15)(17,21,22,29)(18,25,27,26)(19,31,32,43)(20,33,35,34)(23,37,38,28)(24,39,40,41)(30,44,46,45)(36,42,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6931698);