Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C22⋊C4×D8 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15)(3,16,20,46)(4,21)(5,26,28,6)(7,14,9,32)(10,34,37,18)(11,31)(12,13,30,27)(17,47)(19,45,33,43)(22,55)(23,40)(24,42)(25,41)(29,36,58,38)(35,57)(39,54)(44,50)(48,52)(49,62)(51,63)(53,56)(59,61)(60,64);(1,3)(2,10)(4,17)(5,16)(6,29)(7,19)(8,20)(9,33)(11,35)(12,18)(13,43)(14,36)(15,37)(21,52)(22,44)(23,57)(24,49)(25,51)(26,58)(27,45)(28,46)(30,34)(31,50)(32,38)(39,48)(40,47)(41,61)(42,60)(53,64)(54,62)(55,59)(56,63);(1,4)(2,11)(3,17)(5,22)(6,23)(7,31)(8,25)(9,24)(10,35)(12,39)(13,40)(14,41)(15,42)(16,44)(18,48)(19,50)(20,51)(21,32)(26,53)(27,56)(28,54)(29,57)(30,55)(33,49)(34,59)(36,61)(37,60)(38,52)(43,47)(45,63)(46,62)(58,64);(1,5)(2,12)(3,10)(4,22)(6,14)(7,27)(8,28)(9,13)(11,39)(15,30)(16,29)(17,35)(18,43)(19,36)(20,37)(21,53)(23,41)(24,40)(25,54)(26,32)(31,56)(33,38)(34,45)(42,55)(44,57)(46,58)(47,48)(49,52)(50,61)(51,60)(59,63)(62,64)Polycyclic presentation:
2^9|a^2=g,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225388, #72225566, #72225570, … |
| maximal normal subgroup of | #72225388, #72225566, #72225570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404653);