Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238001 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2398 |
| prime radical | #3150 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
530 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 530 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:
(9,10)(11,20,23,33)(12,17)(13,18)(14,28)(15,19)(16,24,30,21)(22,34,37,39)(25,31)(26,32)(27,38)(29,36,40,35);(1,2)(3,5)(4,7)(6,8)(11,21,23,24)(12,25)(16,20,30,33)(17,31)(22,36,37,35)(26,27)(29,39,40,34)(32,38);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,21)(14,24)(15,23)(17,29)(18,33)(19,30)(20,28)(25,37)(26,35)(27,36)(31,40)(32,39)(34,38);(9,12,13,26,15,25,14,27)(10,17,18,32,19,31,28,38)(11,22,21,35,23,37,24,36)(16,29,33,39,30,40,20,34)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ef,f^2=h,g^2=hi,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=dh,g^b=gi,e^c=eh,f^c=fh,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307206, #72307217, #72307226, … |
| maximal normal subgroup of | #72307206, #72307217, #72307226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151645);