Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,20,15,22,31,19,16,21,29,37,12,18,13,30)(11,23,26,35,24,38,25,36,28,39,40,17,27,34,32,33);(2,5)(3,8)(6,7)(9,11)(10,17)(12,24)(13,26)(14,32)(15,27)(16,28)(18,33)(19,34)(20,23)(21,35)(22,36)(25,29)(30,38)(31,40)(37,39);(9,12,16,15)(10,18,21,22)(11,24,28,27)(13,29,31,14)(17,33,35,36)(19,20,30,37)(23,38,39,34)(25,40,32,26)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#124066, #72213051, #72213794, … |
| maximal normal subgroup of | C2×#124066, #72213051, #72213794, … |
| maximal quotient of | #72211371, #72211373, C2×#124066 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31262);