Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246927 |
| 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 | |
1442 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1442 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)(11,20)(12,16)(13,17)(14,19)(15,22)(18,27)(21,36)(23,34)(24,31)(25,33)(26,29)(28,39)(30,37)(32,40)(35,38);(3,7)(5,8)(9,11)(10,15)(12,21)(13,26)(14,23)(16,30)(17,20)(18,34)(19,29)(22,27)(24,39)(25,38)(28,33)(31,36)(32,35)(37,40);(1,2)(3,8)(4,6)(5,7)(9,12,14,25)(10,16,19,33)(11,21,23,38)(13,24,27,40)(15,28,29,30)(17,31,18,32)(20,35,34,36)(22,37,26,39);(3,7)(5,8)(10,17)(13,27)(15,20)(16,31)(18,19)(22,26)(24,40)(28,35)(29,34)(30,36)(32,33)(37,39)Polycyclic presentation:
2^9|c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281791, #72281986, #72285119, … |
| maximal normal subgroup of | #72281791, #72281986, #72285119, … |
| maximal quotient of | #72282146, #72339617 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154124);