Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C4⋊C4 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,13)(3,10)(8,15,16,14)(11,12)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(1,2,5,7,6,9,4,13)(3,8,12,14,10,16,11,15)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3,4,11,6,10,5,12)(2,8,13,15,9,16,7,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,e^b=eh,f^b=fh,d^c=dhi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72321848, #72321883, #72321903, … |
| maximal normal subgroup of | #72321848, #72321883, #72321903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90770);