Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
758 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 758 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,16)(4,21)(5,15)(6,30)(7,13)(8,11)(9,17)(10,18)(12,25)(14,23)(19,50)(20,44)(22,46)(24,33)(26,53)(27,52)(28,37)(29,40)(31,41)(32,45)(34,49)(35,55)(36,54)(38,51)(39,59)(42,57)(43,58)(47,48)(56,64)(60,62)(61,63);(1,3)(2,9)(4,22)(5,18)(6,31)(7,28)(8,20)(10,33)(11,34)(12,41)(13,35)(14,46)(15,47)(16,21)(17,25)(19,23)(24,54)(26,60)(27,56)(29,51)(30,36)(32,61)(37,53)(38,43)(39,63)(40,64)(42,50)(44,52)(45,62)(48,58)(49,59)(55,57);(1,3)(2,10)(4,17)(5,28)(6,19)(7,18)(8,20)(9,33)(11,37)(12,36)(13,44)(14,38)(15,16)(21,47)(22,25)(23,31)(24,55)(26,48)(27,49)(29,61)(30,41)(32,51)(34,53)(35,52)(39,62)(40,50)(42,64)(43,46)(45,63)(54,57)(56,59)(58,60);(2,11)(9,34)(10,37)(12,39)(13,15)(14,40)(16,44)(21,52)(30,45)(33,53)(35,47)(36,62)(38,50)(41,63)(42,43)(46,64)Polycyclic presentation:
2^9|e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224989, #72224995, #72224996, … |
| maximal normal subgroup of | #72224989, #72224995, #72224996, … |
| maximal quotient of | #72227085, C2×#512849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420045);