Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142616 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9946 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 64 points:
(1,2)(3,14)(4,20)(5,9)(6,29)(7,12)(8,15)(10,22)(11,27)(13,23)(16,43)(17,45)(18,38)(19,46)(21,34)(24,39)(25,52)(26,44)(28,36)(30,59)(31,53)(32,49)(33,47)(35,54)(37,56)(40,58)(41,55)(42,48)(50,60)(51,63)(57,61)(62,64);(1,3)(2,8)(4,21)(5,26)(6,18)(7,19)(9,31)(10,38)(11,42)(12,32)(13,34)(14,20)(15,22)(16,27)(17,23)(24,57)(25,47)(28,50)(29,33)(30,36)(35,43)(37,61)(39,44)(40,60)(41,45)(46,52)(48,58)(49,55)(51,56)(53,54)(59,64)(62,63);(2,9)(4,22)(5,27)(6,23)(8,30)(10,39)(11,13)(12,36)(14,43)(15,47)(16,26)(17,18)(20,35)(21,49)(24,58)(25,55)(28,56)(29,37)(31,32)(33,42)(34,61)(38,44)(40,41)(45,60)(46,63)(50,51)(52,62)(53,59)Polycyclic presentation:
2^9|d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72282196 |
| maximal normal subgroup of | #72282196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49812);