Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1415 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,25)(13,16)(14,19)(15,20)(17,23)(18,26)(22,31)(24,30)(27,34)(28,33)(29,32)(35,36)(37,39)(38,40);(3,7)(5,8)(10,16)(13,14)(17,21)(18,25)(19,20)(23,29)(26,28)(30,32)(31,35)(33,34)(36,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,24)(10,17,20,32)(12,22,27,38)(13,23,14,29)(16,30,19,21)(18,31,33,40)(25,39,34,35)(26,36,28,37);(3,7)(5,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,25)(17,31)(19,34)(20,33)(21,35)(23,37)(24,38)(29,36)(30,39)(32,40)Polycyclic presentation:
2^9|c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72337681, #72337687, #72337690, … |
| maximal normal subgroup of | #72337681, #72337687, #72337690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154185);