Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4⋊#95 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(19,27)(23,31)(26,32)(28,29);(3,10)(6,14)(9,15)(11,16)(17,19,20,28,22,27,21,29)(18,23,24,26,25,31,30,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=chi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72211432, C2×#109018, #72213016, … |
| maximal normal subgroup of | #72211432, C2×#109018, #72213016, … |
| maximal quotient of | C2×#109018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16214);