Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,8)(9,10)(11,19,21,15)(12,23,24,16)(13,18)(14,17)(20,27)(22,32,35,28)(25,36,37,29)(26,30)(31,34)(33,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,22,14,21,26,35)(10,15,18,28,17,19,30,32)(12,20,25,33,24,34,37,40)(16,27,29,38,23,31,36,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,20)(13,25)(14,24)(15,27)(17,23)(18,29)(19,31)(21,34)(22,33)(26,37)(28,38)(30,36)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=ei,f^b=fh,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313677, #72313680, #72313702, … |
| maximal normal subgroup of | #72313677, #72313680, #72313702, … |
| maximal quotient of | #72314119, #72314249, C2×#106079 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13275);