Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C2 | C2×D4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,22)(5,23)(6,11)(7,31)(8,13)(9,14)(10,19)(12,30)(16,48)(17,51)(18,41)(20,56)(21,47)(24,58)(25,27)(26,60)(28,59)(29,36)(32,44)(33,42)(34,43)(35,40)(37,61)(38,53)(39,55)(45,49)(46,52)(50,57)(54,64)(62,63);(1,3)(2,10)(4,16)(5,26)(6,18)(7,32)(8,20)(9,21)(11,38)(12,41)(13,44)(14,39)(15,31)(17,28)(19,30)(22,57)(23,49)(24,37)(25,51)(27,54)(29,61)(33,45)(34,55)(35,53)(36,46)(40,47)(42,50)(43,56)(48,58)(52,62)(59,60)(63,64);(1,4)(2,5)(3,16)(6,23)(7,24)(8,36)(9,25)(10,26)(11,27)(12,28)(13,22)(14,29)(15,45)(17,41)(18,49)(19,50)(20,46)(21,51)(30,42)(31,33)(32,37)(34,63)(35,59)(38,54)(39,61)(40,62)(43,58)(44,57)(47,52)(48,56)(53,60)(55,64);(1,5)(2,4)(3,17)(6,27)(7,33)(8,22)(9,29)(10,37)(11,23)(12,42)(13,36)(14,25)(15,46)(16,41)(18,52)(19,54)(20,45)(21,48)(24,31)(26,32)(28,30)(34,58)(35,62)(38,50)(39,60)(40,59)(43,63)(44,64)(47,49)(51,56)(53,61)(55,57);(1,6,9,8)(2,11,14,13)(3,18,21,20)(4,23,25,36)(5,27,29,22)(7,30,35,34)(10,38,39,44)(12,40,43,31)(15,41,47,56)(16,49,51,46)(17,52,48,45)(19,53,55,32)(24,42,59,63)(26,54,61,57)(28,62,58,33)(37,50,60,64)Polycyclic presentation:
2^9|e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,h^a=hi,d^b=df,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530851);