Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10320 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | #32948 | 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 | |
566 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,14,18,38)(4,8,22,30)(5,23,25,11)(6,27,20,13)(9,24,34,16)(10,36,31,19)(15,40,46,59)(17,48,44,42)(21,51,53,29)(26,39,50,55)(28,52,32,45)(33,60,58,64)(35,47,57,56)(37,49,54,62)(41,61,63,43);(2,8)(4,19)(6,20)(9,32)(10,13)(11,29)(12,30)(14,40)(15,43)(16,24)(17,44)(21,49)(22,36)(23,51)(26,50)(27,31)(28,34)(33,57)(35,58)(37,39)(38,59)(41,42)(45,60)(46,61)(48,63)(52,64)(53,62)(54,55);(1,3)(2,9)(4,15)(5,24)(6,17)(7,18)(8,28)(10,33)(11,38)(12,34)(13,35)(14,23)(16,25)(19,43)(20,44)(21,52)(22,46)(26,56)(27,57)(29,59)(30,32)(31,58)(36,61)(37,63)(39,48)(40,51)(41,54)(42,55)(45,53)(47,50)(49,60)(62,64)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,e^c=ei,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72281308 |
| maximal normal subgroup of | #72281308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51438);