Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,31,45,29,34,30,35)(27,36,46,48,38,32,39,52)(28,40,44,33,42,51,43,50)(37,53,55,49,41,56,54,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27)(26,32)(28,41)(29,38)(30,39)(31,46)(33,49)(34,36)(35,48)(37,42)(40,53)(43,55)(44,54)(45,52)(47,50)(51,56);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,29,42)(26,33,34,50)(27,37,38,41)(30,44,31,43)(32,47,36,49)(35,51,45,40)(39,55,46,54)(48,53,52,56);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(26,34)(27,38)(33,50)(35,45)(37,41)(39,46)(40,51)(54,55);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(26,34)(32,36)(33,50)(35,45)(40,51)(47,49)(48,52)(53,56)Polycyclic presentation:
2^9|a^2=gh,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900605);