Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573703 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4374 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)(3,16)(4,11)(5,27)(6,13)(7,35)(8,14)(9,15)(10,20)(12,31)(17,42)(18,51)(19,23)(21,48)(22,41)(24,33)(25,34)(26,37)(28,61)(29,59)(30,44)(32,40)(36,45)(38,46)(39,57)(43,47)(49,55)(50,53)(52,56)(54,60)(58,63)(62,64);(1,3)(2,10)(4,23)(5,18)(6,32)(7,22)(8,20)(9,21)(11,42)(12,39)(13,46)(14,48)(15,41)(16,35)(17,26)(19,34)(24,58)(25,40)(27,60)(28,62)(29,56)(30,57)(31,55)(33,53)(36,54)(37,38)(43,64)(44,51)(45,49)(47,52)(50,59)(61,63);(1,4,9,13)(2,11,15,6)(3,17,21,40)(5,28,31,59)(7,26,8,25)(10,38,41,19)(12,29,27,61)(14,34,35,37)(16,32,48,42)(18,52,55,58)(20,23,22,46)(24,36,47,30)(33,45,43,44)(39,53,60,64)(49,56,51,63)(50,57,62,54);(1,5)(2,12)(3,18)(4,24)(6,33)(7,30)(8,36)(9,31)(10,39)(11,43)(13,47)(14,44)(15,27)(16,49)(17,50)(19,56)(20,54)(21,55)(22,57)(23,58)(25,28)(26,59)(29,34)(32,53)(35,45)(37,61)(38,63)(40,62)(41,60)(42,64)(46,52)(48,51);(1,6,9,11)(2,13,15,4)(3,19,21,38)(5,29,31,61)(7,34,8,37)(10,40,41,17)(12,28,27,59)(14,26,35,25)(16,23,48,46)(18,53,55,64)(20,32,22,42)(24,44,47,45)(30,43,36,33)(39,52,60,58)(49,62,51,50)(54,63,57,56)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72341326, #72342022, #72342045 |
| maximal normal subgroup of | #72341326, #72342045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480904);