Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36,32,37,30,46)(27,33,40,49,41,50,39,54)(28,42,43,51,29,35,45,34)(38,48,53,56,44,52,55,47);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,33)(28,38)(29,44)(30,39)(31,40)(32,41)(34,47)(35,52)(36,49)(37,50)(42,48)(43,53)(45,55)(46,54)(51,56);(2,5)(6,7)(25,28,32,29)(26,34,37,51)(27,38,41,44)(30,43,31,45)(33,47,50,56)(35,46,42,36)(39,53,40,55)(48,49,52,54);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,32,28)(26,35,37,42)(27,38,41,44)(30,43,31,45)(33,48,50,52)(34,46,51,36)(39,55,40,53)(47,49,56,54);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(22,23)(25,30,32,31)(26,36,37,46)(27,39,41,40)(28,43,29,45)(33,49,50,54)(34,35,51,42)(38,53,44,55)(47,52,56,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6709675);