Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232766 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2397 |
| prime radical | #2397 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
700 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10,15,28,14,19,13,18)(11,20,27,16,23,33,22,30)(12,24,29,32,26,17,25,39)(21,35,40,38,37,34,36,31);(2,6)(3,4)(7,8)(9,11)(10,16)(12,21)(13,27)(14,23)(15,22)(17,31)(18,33)(19,30)(20,28)(24,38)(25,40)(26,37)(29,36)(32,35)(34,39);(2,7)(3,4)(6,8)(16,30)(20,33)(31,38)(34,35);(2,8)(6,7)(9,12,13,25,14,26,15,29)(10,17,18,32,19,24,28,39)(11,21,27,40,23,37,22,36)(16,31,33,35,30,38,20,34)Polycyclic presentation:
2^9|a^2=eh,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283833, C2×#244123, #72307201, … |
| maximal normal subgroup of | #72283833, C2×#244123, #72307201, … |
| maximal quotient of | C2×#244123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151319);