Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238001 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,24,26,35)(13,19)(14,20)(15,18)(17,27,34,28)(21,33,37,40)(22,32)(23,31)(25,39,38,30)(29,36);(9,11)(10,16)(12,25,26,38)(13,29)(14,22)(15,23)(17,33,34,40)(18,31)(19,36)(20,32)(21,28,37,27)(24,30,35,39);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,34)(19,35)(20,24)(22,38)(23,37)(25,29)(31,39)(32,40)(33,36);(10,18)(16,31)(17,34)(19,20)(24,35)(30,39)(32,36)(33,40)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279774, #72279782, #72279882, … |
| maximal normal subgroup of | #72279774, #72279782, #72279882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145197);