Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33717 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,7,10,13,19,15,11,14,8,16,18,20,17,12);(6,12,14,15)(7,13,16,20)(8,17)(9,19)(10,18);(1,2,3,4);(5,7,8,13,10,15,17,14,11,16,9,20,18,12,19,6)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227103, D4×AGL(1,ℤ/16), #72227173, … |
| maximal normal subgroup of | #72227103, D4×AGL(1,ℤ/16), #72227173, … |
| maximal quotient of | #72227100, #72227102, #72227103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405774);