Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,16,29)(4,18,19,10)(5,20,22,11)(6,9)(7,12)(8,15,26,31)(14,24,21,27)(17,25,30,32)(23,28);(1,3,7,17)(2,8,12,27)(4,15,19,14)(5,21,22,31)(6,16,23,30)(9,24,28,26)(10,13,18,25)(11,29,20,32);(2,9)(8,24)(10,11)(12,28)(13,29)(18,20)(25,32)(26,27);(1,4,6,5)(2,10,9,11)(3,14,16,31)(7,19,23,22)(8,25,24,32)(12,18,28,20)(13,26,29,27)(15,30,21,17)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284058, #72284122, #72284152, … |
| maximal normal subgroup of | #72284058, #72284122, #72284152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389759);