Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C23 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,12,19,14,20,15,13,17,22,32,33,16,21,31,34)(11,23,25,39,27,35,40,38,24,37,30,26,29,18,28,36);(2,5)(6,8)(9,11,13,26,15,28,21,37,16,29,19,38,12,30,22,35,17,24,34,39,31,40,20,23,14,27,33,36,32,25,10,18);(11,24)(18,35)(23,37)(25,30)(26,39)(27,29)(28,40)(36,38)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bf,e^a=ei,c^b=ci,d^b=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72210680, #72225138, #72225141, … |
| maximal normal subgroup of | #72210680, #72225138, #72225141, … |
| maximal quotient of | #72225137, C2×#149502, #72234237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56698);