Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | C2×#1393 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9)(3,12,14,29)(5,10,19,27)(6,11,17,28)(7,21,23,31)(8,18,25,20)(13,24,30,26)(15,22,16,32);(1,3)(2,7)(4,16,17,21)(5,15)(6,13)(8,24)(9,26,28,29)(10,12)(11,22)(14,19,30,18)(20,31)(23,25,32,27);(3,13)(7,22)(12,24)(14,30)(15,31)(16,21)(23,32)(26,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=hi,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=di,e^b=egh,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284737, #72284759, #72284783, … |
| maximal normal subgroup of | #72284737, #72284759, #72284783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59060);