Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,21,24,35)(12,17)(13,27,29,18)(14,19)(15,20)(16,23)(22,37,25,38)(26,36,30,34)(28,33)(31,39)(32,40);(1,3)(2,5,6,7)(4,8)(9,11,14,25,15,24,12,22)(10,16,19,32,20,33,17,31)(13,28,30,39,29,23,26,40)(18,35,36,37,27,21,34,38);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,29,30)(16,31,33,32)(18,34,27,36)(21,37,35,38)(23,39,28,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,d^b=dh,e^b=egi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307758, #72308082, #72308105, … |
| maximal normal subgroup of | #72307758, #72308082, #72308105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35232);