Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,20,28,22,27,21,29)(18,23,24,26,25,31,30,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,14)(10,15)(12,16);(2,8)(6,13)(7,14)(12,16)(19,27)(23,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=eh,d^c=dhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72322104, #72322139, #72322157 |
| maximal normal subgroup of | #72322104, #72322139, #72322157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251585);