Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,10)(7,14)(8,16)(11,17)(12,18)(13,19)(15,20);(1,3,4,2)(5,6)(7,11)(8,17)(9,12)(10,14)(13,16)(15,19)(18,20);(7,15)(10,18)(11,19)(14,20);(5,7,8,14,9,15,16,20)(6,11,17,10,12,19,13,18)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=fh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72221563, C4×#33688, #72258315, … |
| maximal normal subgroup of | C4×#33688, #72258315, #72258327 |
| maximal quotient of | #72221563, #72242485 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402677);