Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4241 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)(3,13,17,8)(4,9)(5,21,23,10)(6,11)(7,12)(14,26,40,29)(15,27)(16,37,44,28)(18,45,47,30)(19,31)(20,32)(22,48,50,33)(24,34)(25,35)(36,42)(38,55)(39,52,43,54)(41,53)(46,57,49,56)(51,60)(58,62)(59,64)(61,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,29)(10,27)(11,28)(12,13)(18,38)(19,39)(20,40)(21,36)(22,41)(23,42)(24,43)(25,44)(26,32)(30,55)(31,54)(33,53)(34,52)(35,37)(45,51)(46,59)(47,60)(48,58)(49,61)(50,62)(56,64)(57,63);(1,4,7,20)(2,9,12,32)(3,14,17,40)(5,18,23,47)(6,24,25,19)(8,26,13,29)(10,30,21,45)(11,34,35,31)(15,38,42,60)(16,43,44,39)(22,49,50,46)(27,51,36,55)(28,54,37,52)(33,57,48,56)(41,61,62,59)(53,64,58,63);(1,5)(2,10)(3,15)(4,18)(6,22)(7,23)(8,27)(9,30)(11,33)(12,21)(13,36)(14,38)(16,41)(17,42)(19,46)(20,47)(24,49)(25,50)(26,51)(28,53)(29,55)(31,56)(32,45)(34,57)(35,48)(37,58)(39,59)(40,60)(43,61)(44,62)(52,63)(54,64);(1,6,7,25)(2,11,12,35)(3,16,17,44)(4,19,20,24)(5,22,23,50)(8,28,13,37)(9,31,32,34)(10,33,21,48)(14,39,40,43)(15,41,42,62)(18,46,47,49)(26,52,29,54)(27,53,36,58)(30,56,45,57)(38,59,60,61)(51,63,55,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,e^c=ei,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72310866, #72310890, #72310907 |
| maximal normal subgroup of | #72310866, #72310890, #72310907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281385);