Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,30)(27,36)(29,44)(31,40)(32,38)(33,48)(34,42)(35,43)(37,47)(39,46)(41,45);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,37)(26,27)(28,40)(29,33)(34,35)(45,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,31,36,47)(18,24,27,28,23,37,26,40)(20,33,35,45,32,29,34,46)(25,41,42,48,38,39,43,44);(2,6)(7,8)(11,16)(13,15)(17,20,22,32)(18,25,23,38)(19,29,31,33)(21,35,36,34)(24,39,37,41)(26,43,27,42)(28,44,40,48)(30,46,47,45);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,36)(18,26,23,27)(19,30,31,47)(20,34,32,35)(24,40,37,28)(25,42,38,43)(29,45,33,46)(39,44,41,48)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=i,e^2=f,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=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6935545);