Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | |
1189 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,34)(27,31,36,44)(28,38,39,53)(29,41,42,33)(32,46,47,40)(35,48,50,56)(37,52,51,45)(43,54,55,49);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,36)(26,31,34,44)(28,35,39,50)(29,37,42,51)(32,43,47,55)(33,45,41,52)(38,48,53,56)(40,49,46,54);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,29,40,30,39,42,46)(26,32,33,38,34,47,41,53)(27,35,37,49,36,50,51,54)(31,43,45,48,44,55,52,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,36)(28,39)(31,44)(32,47)(37,51)(38,53)(40,46)(45,52);(1,3,4,8)(2,6,5,7)(11,21)(15,18)(19,20)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6712038);