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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(10,14)(11,17)(12,20)(22,24)(25,26,30,36,31,35,45,46)(27,37,39,48,28,33,41,32)(29,34,43,52,44,51,53,56)(38,50,42,55,40,49,54,47);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27)(26,32)(28,31)(29,42)(30,41)(33,36)(34,50)(35,48)(37,46)(38,43)(39,45)(40,53)(44,54)(47,52)(49,51)(55,56);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,27)(26,33,35,37)(29,40,44,38)(30,41,45,39)(32,46,48,36)(34,49,51,50)(42,43,54,53)(47,56,55,52);(10,14)(11,17)(16,21)(19,23)(25,29,30,43,31,44,45,53)(26,34,36,52,35,51,46,56)(27,38,39,42,28,40,41,54)(32,47,37,50,48,55,33,49);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(26,35)(27,28)(34,51)(36,46)(38,40)(39,41)(42,54)(52,56)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gi,d^2=fhi,e^2=hi,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107096);