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 |
| prime radical | #25799 | C2 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 562 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:
(1,2,7,12)(3,14,19,49)(4,20,25,54)(5,9,28,37)(6,29,23,13)(8,30,33,15)(10,39,43,22)(11,44,38,27)(16,46,53,64)(17,48,18,40)(21,35,50,34)(24,41,58,42)(26,47,52,59)(31,55,32,61)(36,62,63,56)(45,57,60,51);(1,3)(2,8)(4,21)(5,26)(6,18)(7,19)(9,32)(10,40)(11,45)(12,33)(13,35)(14,20)(15,22)(16,27)(17,23)(24,57)(25,50)(28,52)(29,34)(30,39)(31,37)(36,46)(38,60)(41,47)(42,59)(43,48)(44,53)(49,54)(51,58)(55,56)(61,62)(63,64);(2,9)(4,22)(5,27)(6,23)(8,31)(10,41)(11,13)(12,37)(14,46)(15,50)(16,26)(17,18)(20,36)(21,30)(24,58)(25,39)(28,44)(29,38)(32,33)(34,45)(35,60)(40,47)(42,43)(48,59)(49,64)(52,53)(54,63)(55,61)Polycyclic presentation:
2^9|a^2=i,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, 49814);