Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,29,32,42)(20,33,34,26)(21,28,23,27)(25,37,38,31)(30,44,45,43)(35,40,36,41)(39,47,48,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(20,34)(25,38)(27,28)(29,42)(30,45)(35,36)(46,47);(11,15)(13,16)(17,19,22,32)(18,25,24,38)(20,30,34,45)(21,31,23,37)(26,39,33,48)(27,29,28,42)(35,46,36,47)(40,44,41,43);(3,8)(5,7)(10,14)(11,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,33)(25,39)(27,41)(28,40)(29,43)(31,47)(32,45)(37,46)(38,48)(42,44);(2,6)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,28)(19,31,32,37)(20,35,34,36)(25,29,38,42)(26,40,33,41)(30,46,45,47)(39,44,48,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107561);