Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62862 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2290 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 2290 conjugacy classes of subgroups — too many to draw. Only its 188 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)(9,10,14,19)(11,16,22,30)(12,17,24,32)(13,15,25,18)(20,33,35,28)(21,26,36,29)(23,27,37,31)(34,39,40,38);(3,7)(5,8)(10,15)(13,25)(16,26)(17,27)(18,19)(21,36)(23,37)(28,38)(29,30)(31,32)(33,39)(34,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,26)(17,28)(18,29)(19,30)(23,34)(24,35)(25,36)(27,38)(31,39)(32,33)(37,40);(9,12)(10,17)(11,20)(13,23)(14,24)(15,27)(16,28)(18,31)(19,32)(21,34)(22,35)(25,37)(26,38)(29,39)(30,33)(36,40)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gh,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72226694, #72283973, #72284167, … |
| maximal normal subgroup of | #72226694, #72283973, #72284167, … |
| maximal quotient of | #72219584 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254360);